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In the above example, the algebraic cofactor of ",[17,3495,3497,3510],{"className":3496,"translate":21},[20],[17,3498,3500],{"className":3499},[25],[27,3501,3502],{"xmlns":29},[31,3503,3504,3508],{},[34,3505,3506],{},[37,3507,172],{},[41,3509,172],{"encoding":43},[17,3511,3513],{"className":3512,"ariaHidden":48},[47],[17,3514,3516,3519],{"className":3515},[52],[17,3517],{"className":3518,"style":123},[56],[17,3520,172],{"className":3521,"style":187},[61,62]," is",[17,3524,3526],{"className":3525,"translate":21},[807],[17,3527,3529,3604],{"className":3528,"translate":21},[20],[17,3530,3532],{"className":3531},[25],[27,3533,3534],{"xmlns":29,"display":816},[31,3535,3536,3601],{},[34,3537,3538,3540,3542,3545,3585,3591,3593,3595],{},[204,3539,622],{"stretchy":621},[204,3541,627],{},[408,3543,3544],{},"1",[403,3546,3547,3549],{},[204,3548,632],{"stretchy":621},[34,3550,3551,3553,3555,3558,3560,3562,3565,3567,3569,3571,3573,3575,3578,3580,3583],{},[204,3552,622],{"stretchy":621},[408,3554,3544],{},[204,3556,3557],{},"+",[408,3559,410],{},[204,3561,3557],{},[408,3563,3564],{},"4",[204,3566,632],{"stretchy":621},[204,3568,3557],{},[204,3570,622],{"stretchy":621},[408,3572,410],{},[204,3574,3557],{},[408,3576,3577],{},"3",[204,3579,3557],{},[408,3581,3582],{},"5",[204,3584,632],{"stretchy":621},[403,3586,3587,3589],{},[37,3588,172],{},[204,3590,695],{"mathvariant":693,"lspace":694,"rspace":694},[204,3592,825],{},[204,3594,627],{},[403,3596,3597,3599],{},[37,3598,172],{},[204,3600,695],{"mathvariant":693,"lspace":694,"rspace":694},[41,3602,3603],{"encoding":43},"(-1)^{(1+2+4)+(2+3+5)}M'=-M'",[17,3605,3607,3739],{"className":3606,"ariaHidden":48},[47],[17,3608,3610,3614,3617,3620,3623,3698,3730,3733,3736],{"className":3609},[52],[17,3611],{"className":3612,"style":3613},[56],"height:1.188em;vertical-align:-0.25em;",[17,3615,622],{"className":3616},[649],[17,3618,627],{"className":3619},[61],[17,3621,3544],{"className":3622},[61],[17,3624,3626,3629],{"className":3625},[674],[17,3627,632],{"className":3628},[674],[17,3630,3632],{"className":3631},[433],[17,3633,3635],{"className":3634},[437],[17,3636,3638],{"className":3637},[441],[17,3639,3642],{"className":3640,"style":3641},[445],"height:0.938em;",[17,3643,3644,3647],{"style":3110},[17,3645],{"className":3646,"style":453},[452],[17,3648,3650],{"className":3649},[457,458,459,460],[17,3651,3653,3656,3659,3662,3665,3668,3671,3674,3677,3680,3683,3686,3689,3692,3695],{"className":3652},[61,460],[17,3654,622],{"className":3655},[649,460],[17,3657,3544],{"className":3658},[61,460],[17,3660,3557],{"className":3661},[233,460],[17,3663,410],{"className":3664},[61,460],[17,3666,3557],{"className":3667},[233,460],[17,3669,3564],{"className":3670},[61,460],[17,3672,632],{"className":3673},[674,460],[17,3675,3557],{"className":3676},[233,460],[17,3678,622],{"className":3679},[649,460],[17,3681,410],{"className":3682},[61,460],[17,3684,3557],{"className":3685},[233,460],[17,3687,3577],{"className":3688},[61,460],[17,3690,3557],{"className":3691},[233,460],[17,3693,3582],{"className":3694},[61,460],[17,3696,632],{"className":3697},[674,460],[17,3699,3701,3704],{"className":3700},[61],[17,3702,172],{"className":3703,"style":187},[61,62],[17,3705,3707],{"className":3706},[433],[17,3708,3710],{"className":3709},[437],[17,3711,3713],{"className":3712},[441],[17,3714,3716],{"className":3715,"style":3089},[445],[17,3717,3718,3721],{"style":3110},[17,3719],{"className":3720,"style":453},[452],[17,3722,3724],{"className":3723},[457,458,459,460],[17,3725,3727],{"className":3726},[61,460],[17,3728,695],{"className":3729},[61,460],[17,3731],{"className":3732,"style":372},[228],[17,3734,825],{"className":3735},[376],[17,3737],{"className":3738,"style":372},[228],[17,3740,3742,3746,3749],{"className":3741},[52],[17,3743],{"className":3744,"style":3745},[56],"height:0.8852em;vertical-align:-0.0833em;",[17,3747,627],{"className":3748},[61],[17,3750,3752,3755],{"className":3751},[61],[17,3753,172],{"className":3754,"style":187},[61,62],[17,3756,3758],{"className":3757},[433],[17,3759,3761],{"className":3760},[437],[17,3762,3764],{"className":3763},[441],[17,3765,3767],{"className":3766,"style":3089},[445],[17,3768,3769,3772],{"style":3110},[17,3770],{"className":3771,"style":453},[452],[17,3773,3775],{"className":3774},[457,458,459,460],[17,3776,3778],{"className":3777},[61,460],[17,3779,695],{"className":3780},[61,460],[10,3782,3783,3786,3787,3815,3816,3844,3845,3873,3874,4290,4291,4352,4353,4381,4382,4828,4829,798],{},[13,3784,3785],{},"Definition (Algebraic Cofactor):"," Let ",[17,3788,3790,3803],{"className":3789,"translate":21},[20],[17,3791,3793],{"className":3792},[25],[27,3794,3795],{"xmlns":29},[31,3796,3797,3801],{},[34,3798,3799],{},[37,3800,172],{},[41,3802,172],{"encoding":43},[17,3804,3806],{"className":3805,"ariaHidden":48},[47],[17,3807,3809,3812],{"className":3808},[52],[17,3810],{"className":3811,"style":123},[56],[17,3813,172],{"className":3814,"style":187},[61,62]," be a ",[17,3817,3819,3832],{"className":3818,"translate":21},[20],[17,3820,3822],{"className":3821},[25],[27,3823,3824],{"xmlns":29},[31,3825,3826,3830],{},[34,3827,3828],{},[37,3829,39],{},[41,3831,39],{"encoding":43},[17,3833,3835],{"className":3834,"ariaHidden":48},[47],[17,3836,3838,3841],{"className":3837},[52],[17,3839],{"className":3840,"style":57},[56],[17,3842,39],{"className":3843,"style":63},[61,62],"-th order minor of ",[17,3846,3848,3861],{"className":3847,"translate":21},[20],[17,3849,3851],{"className":3850},[25],[27,3852,3853],{"xmlns":29},[31,3854,3855,3859],{},[34,3856,3857],{},[37,3858,111],{},[41,3860,111],{"encoding":43},[17,3862,3864],{"className":3863,"ariaHidden":48},[47],[17,3865,3867,3870],{"className":3866},[52],[17,3868],{"className":3869,"style":123},[56],[17,3871,111],{"className":3872,"style":127},[61,62]," with row and column indices ",[17,3875,3877,3956],{"className":3876,"translate":21},[20],[17,3878,3880],{"className":3879},[25],[27,3881,3882],{"xmlns":29},[31,3883,3884,3953],{},[34,3885,3886,3893,3895,3901,3903,3906,3910,3912,3918,3921,3924,3931,3933,3939,3941,3943,3945,3947],{},[848,3887,3888,3891],{},[37,3889,3890],{},"i",[408,3892,3544],{},[204,3894,1084],{"separator":48},[848,3896,3897,3899],{},[37,3898,3890],{},[408,3900,410],{},[204,3902,1084],{"separator":48},[204,3904,3905],{},"⋯",[3907,3908,3909],"mtext",{}," ",[204,3911,1084],{"separator":48},[848,3913,3914,3916],{},[37,3915,3890],{},[37,3917,39],{},[204,3919,3920],{"separator":48},";",[3907,3922,3923],{}," ",[848,3925,3926,3929],{},[37,3927,3928],{},"j",[408,3930,3544],{},[204,3932,1084],{"separator":48},[848,3934,3935,3937],{},[37,3936,3928],{},[408,3938,410],{},[204,3940,1084],{"separator":48},[204,3942,3905],{},[3907,3944,3909],{},[204,3946,1084],{"separator":48},[848,3948,3949,3951],{},[37,3950,3928],{},[37,3952,39],{},[41,3954,3955],{"encoding":43},"i_{1},i_{2},\\cdots,i_{k}; \\ j_{1},j_{2},\\cdots,j_{k}",[17,3957,3959],{"className":3958,"ariaHidden":48},[47],[17,3960,3962,3966,4009,4012,4015,4058,4061,4064,4067,4070,4073,4076,4079,4123,4126,4129,4132,4177,4180,4183,4226,4229,4232,4235,4238,4241,4244,4247],{"className":3961},[52],[17,3963],{"className":3964,"style":3965},[56],"height:0.854em;vertical-align:-0.1944em;",[17,3967,3969,3972],{"className":3968},[61],[17,3970,3890],{"className":3971},[61,62],[17,3973,3975],{"className":3974},[433],[17,3976,3978,4001],{"className":3977},[437,1313],[17,3979,3981,3998],{"className":3980},[441],[17,3982,3984],{"className":3983,"style":1404},[445],[17,3985,3986,3989],{"style":1407},[17,3987],{"className":3988,"style":453},[452],[17,3990,3992],{"className":3991},[457,458,459,460],[17,3993,3995],{"className":3994},[61,460],[17,3996,3544],{"className":3997},[61,460],[17,3999,1346],{"className":4000},[1345],[17,4002,4004],{"className":4003},[441],[17,4005,4007],{"className":4006,"style":1429},[445],[17,4008],{},[17,4010,1084],{"className":4011},[2410],[17,4013],{"className":4014,"style":2406},[228],[17,4016,4018,4021],{"className":4017},[61],[17,4019,3890],{"className":4020},[61,62],[17,4022,4024],{"className":4023},[433],[17,4025,4027,4050],{"className":4026},[437,1313],[17,4028,4030,4047],{"className":4029},[441],[17,4031,4033],{"className":4032,"style":1404},[445],[17,4034,4035,4038],{"style":1407},[17,4036],{"className":4037,"style":453},[452],[17,4039,4041],{"className":4040},[457,458,459,460],[17,4042,4044],{"className":4043},[61,460],[17,4045,410],{"className":4046},[61,460],[17,4048,1346],{"className":4049},[1345],[17,4051,4053],{"className":4052},[441],[17,4054,4056],{"className":4055,"style":1429},[445],[17,4057],{},[17,4059,1084],{"className":4060},[2410],[17,4062],{"className":4063,"style":2406},[228],[17,4065,3905],{"className":4066},[1301],[17,4068],{"className":4069,"style":2406},[228],[17,4071],{"className":4072,"style":2406},[228],[17,4074,1084],{"className":4075},[2410],[17,4077],{"className":4078,"style":2406},[228],[17,4080,4082,4085],{"className":4081},[61],[17,4083,3890],{"className":4084},[61,62],[17,4086,4088],{"className":4087},[433],[17,4089,4091,4115],{"className":4090},[437,1313],[17,4092,4094,4112],{"className":4093},[441],[17,4095,4098],{"className":4096,"style":4097},[445],"height:0.3361em;",[17,4099,4100,4103],{"style":1407},[17,4101],{"className":4102,"style":453},[452],[17,4104,4106],{"className":4105},[457,458,459,460],[17,4107,4109],{"className":4108},[61,460],[17,4110,39],{"className":4111,"style":63},[61,62,460],[17,4113,1346],{"className":4114},[1345],[17,4116,4118],{"className":4117},[441],[17,4119,4121],{"className":4120,"style":1429},[445],[17,4122],{},[17,4124,3920],{"className":4125},[2410],[17,4127,3923],{"className":4128},[228],[17,4130],{"className":4131,"style":2406},[228],[17,4133,4135,4139],{"className":4134},[61],[17,4136,3928],{"className":4137,"style":4138},[61,62],"margin-right:0.05724em;",[17,4140,4142],{"className":4141},[433],[17,4143,4145,4169],{"className":4144},[437,1313],[17,4146,4148,4166],{"className":4147},[441],[17,4149,4151],{"className":4150,"style":1404},[445],[17,4152,4154,4157],{"style":4153},"top:-2.55em;margin-left:-0.0572em;margin-right:0.05em;",[17,4155],{"className":4156,"style":453},[452],[17,4158,4160],{"className":4159},[457,458,459,460],[17,4161,4163],{"className":4162},[61,460],[17,4164,3544],{"className":4165},[61,460],[17,4167,1346],{"className":4168},[1345],[17,4170,4172],{"className":4171},[441],[17,4173,4175],{"className":4174,"style":1429},[445],[17,4176],{},[17,4178,1084],{"className":4179},[2410],[17,4181],{"className":4182,"style":2406},[228],[17,4184,4186,4189],{"className":4185},[61],[17,4187,3928],{"className":4188,"style":4138},[61,62],[17,4190,4192],{"className":4191},[433],[17,4193,4195,4218],{"className":4194},[437,1313],[17,4196,4198,4215],{"className":4197},[441],[17,4199,4201],{"className":4200,"style":1404},[445],[17,4202,4203,4206],{"style":4153},[17,4204],{"className":4205,"style":453},[452],[17,4207,4209],{"className":4208},[457,458,459,460],[17,4210,4212],{"className":4211},[61,460],[17,4213,410],{"className":4214},[61,460],[17,4216,1346],{"className":4217},[1345],[17,4219,4221],{"className":4220},[441],[17,4222,4224],{"className":4223,"style":1429},[445],[17,4225],{},[17,4227,1084],{"className":4228},[2410],[17,4230],{"className":4231,"style":2406},[228],[17,4233,3905],{"className":4234},[1301],[17,4236],{"className":4237,"style":2406},[228],[17,4239],{"className":4240,"style":2406},[228],[17,4242,1084],{"className":4243},[2410],[17,4245],{"className":4246,"style":2406},[228],[17,4248,4250,4253],{"className":4249},[61],[17,4251,3928],{"className":4252,"style":4138},[61,62],[17,4254,4256],{"className":4255},[433],[17,4257,4259,4282],{"className":4258},[437,1313],[17,4260,4262,4279],{"className":4261},[441],[17,4263,4265],{"className":4264,"style":4097},[445],[17,4266,4267,4270],{"style":4153},[17,4268],{"className":4269,"style":453},[452],[17,4271,4273],{"className":4272},[457,458,459,460],[17,4274,4276],{"className":4275},[61,460],[17,4277,39],{"className":4278,"style":63},[61,62,460],[17,4280,1346],{"className":4281},[1345],[17,4283,4285],{"className":4284},[441],[17,4286,4288],{"className":4287,"style":1429},[445],[17,4289],{},", respectively. The cofactor ",[17,4292,4294,4311],{"className":4293,"translate":21},[20],[17,4295,4297],{"className":4296},[25],[27,4298,4299],{"xmlns":29},[31,4300,4301,4309],{},[34,4302,4303],{},[403,4304,4305,4307],{},[37,4306,172],{},[204,4308,695],{"mathvariant":693,"lspace":694,"rspace":694},[41,4310,698],{"encoding":43},[17,4312,4314],{"className":4313,"ariaHidden":48},[47],[17,4315,4317,4320],{"className":4316},[52],[17,4318],{"className":4319,"style":708},[56],[17,4321,4323,4326],{"className":4322},[61],[17,4324,172],{"className":4325,"style":187},[61,62],[17,4327,4329],{"className":4328},[433],[17,4330,4332],{"className":4331},[437],[17,4333,4335],{"className":4334},[441],[17,4336,4338],{"className":4337,"style":708},[445],[17,4339,4340,4343],{"style":448},[17,4341],{"className":4342,"style":453},[452],[17,4344,4346],{"className":4345},[457,458,459,460],[17,4347,4349],{"className":4348},[61,460],[17,4350,695],{"className":4351},[61,460]," of ",[17,4354,4356,4369],{"className":4355,"translate":21},[20],[17,4357,4359],{"className":4358},[25],[27,4360,4361],{"xmlns":29},[31,4362,4363,4367],{},[34,4364,4365],{},[37,4366,172],{},[41,4368,172],{"encoding":43},[17,4370,4372],{"className":4371,"ariaHidden":48},[47],[17,4373,4375,4378],{"className":4374},[52],[17,4376],{"className":4377,"style":123},[56],[17,4379,172],{"className":4380,"style":187},[61,62]," is multiplied by the sign ",[17,4383,4385,4471],{"className":4384,"translate":21},[20],[17,4386,4388],{"className":4387},[25],[27,4389,4390],{"xmlns":29},[31,4391,4392,4468],{},[34,4393,4394,4396,4398,4400],{},[204,4395,622],{"stretchy":621},[204,4397,627],{},[408,4399,3544],{},[403,4401,4402,4404],{},[204,4403,632],{"stretchy":621},[34,4405,4406,4408,4414,4416,4422,4424,4426,4428,4434,4436,4438,4440,4446,4448,4454,4456,4458,4460,4466],{},[204,4407,622],{"stretchy":621},[848,4409,4410,4412],{},[37,4411,3890],{},[408,4413,3544],{},[204,4415,3557],{},[848,4417,4418,4420],{},[37,4419,3890],{},[408,4421,410],{},[204,4423,3557],{},[204,4425,3905],{},[204,4427,3557],{},[848,4429,4430,4432],{},[37,4431,3890],{},[37,4433,39],{},[204,4435,632],{"stretchy":621},[204,4437,3557],{},[204,4439,622],{"stretchy":621},[848,4441,4442,4444],{},[37,4443,3928],{},[408,4445,3544],{},[204,4447,3557],{},[848,4449,4450,4452],{},[37,4451,3928],{},[408,4453,410],{},[204,4455,3557],{},[204,4457,3905],{},[204,4459,3557],{},[848,4461,4462,4464],{},[37,4463,3928],{},[37,4465,39],{},[204,4467,632],{"stretchy":621},[41,4469,4470],{"encoding":43},"(-1)^{(i_{1}+i_{2}+\\cdots+i_{k})+(j_{1}+j_{2}+\\cdots+j_{k})}",[17,4472,4474],{"className":4473,"ariaHidden":48},[47],[17,4475,4477,4481,4484,4487,4490],{"className":4476},[52],[17,4478],{"className":4479,"style":4480},[56],"height:1.138em;vertical-align:-0.25em;",[17,4482,622],{"className":4483},[649],[17,4485,627],{"className":4486},[61],[17,4488,3544],{"className":4489},[61],[17,4491,4493,4496],{"className":4492},[674],[17,4494,632],{"className":4495},[674],[17,4497,4499],{"className":4498},[433],[17,4500,4502],{"className":4501},[437],[17,4503,4505],{"className":4504},[441],[17,4506,4509],{"className":4507,"style":4508},[445],"height:0.888em;",[17,4510,4511,4514],{"style":448},[17,4512],{"className":4513,"style":453},[452],[17,4515,4517],{"className":4516},[457,458,459,460],[17,4518,4520,4523,4572,4575,4618,4621,4624,4627,4673,4676,4679,4682,4726,4729,4772,4775,4778,4781,4825],{"className":4519},[61,460],[17,4521,622],{"className":4522},[649,460],[17,4524,4526,4529],{"className":4525},[61,460],[17,4527,3890],{"className":4528},[61,62,460],[17,4530,4532],{"className":4531},[433],[17,4533,4535,4563],{"className":4534},[437,1313],[17,4536,4538,4560],{"className":4537},[441],[17,4539,4542],{"className":4540,"style":4541},[445],"height:0.3173em;",[17,4543,4545,4549],{"style":4544},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[17,4546],{"className":4547,"style":4548},[452],"height:2.5em;",[17,4550,4554],{"className":4551},[457,4552,4553,460],"reset-size3","size1",[17,4555,4557],{"className":4556},[61,460],[17,4558,3544],{"className":4559},[61,460],[17,4561,1346],{"className":4562},[1345],[17,4564,4566],{"className":4565},[441],[17,4567,4570],{"className":4568,"style":4569},[445],"height:0.143em;",[17,4571],{},[17,4573,3557],{"className":4574},[233,460],[17,4576,4578,4581],{"className":4577},[61,460],[17,4579,3890],{"className":4580},[61,62,460],[17,4582,4584],{"className":4583},[433],[17,4585,4587,4610],{"className":4586},[437,1313],[17,4588,4590,4607],{"className":4589},[441],[17,4591,4593],{"className":4592,"style":4541},[445],[17,4594,4595,4598],{"style":4544},[17,4596],{"className":4597,"style":4548},[452],[17,4599,4601],{"className":4600},[457,4552,4553,460],[17,4602,4604],{"className":4603},[61,460],[17,4605,410],{"className":4606},[61,460],[17,4608,1346],{"className":4609},[1345],[17,4611,4613],{"className":4612},[441],[17,4614,4616],{"className":4615,"style":4569},[445],[17,4617],{},[17,4619,3557],{"className":4620},[233,460],[17,4622,3905],{"className":4623},[1301,460],[17,4625,3557],{"className":4626},[233,460],[17,4628,4630,4633],{"className":4629},[61,460],[17,4631,3890],{"className":4632},[61,62,460],[17,4634,4636],{"className":4635},[433],[17,4637,4639,4664],{"className":4638},[437,1313],[17,4640,4642,4661],{"className":4641},[441],[17,4643,4646],{"className":4644,"style":4645},[445],"height:0.3448em;",[17,4647,4649,4652],{"style":4648},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[17,4650],{"className":4651,"style":4548},[452],[17,4653,4655],{"className":4654},[457,4552,4553,460],[17,4656,4658],{"className":4657},[61,460],[17,4659,39],{"className":4660,"style":63},[61,62,460],[17,4662,1346],{"className":4663},[1345],[17,4665,4667],{"className":4666},[441],[17,4668,4671],{"className":4669,"style":4670},[445],"height:0.1512em;",[17,4672],{},[17,4674,632],{"className":4675},[674,460],[17,4677,3557],{"className":4678},[233,460],[17,4680,622],{"className":4681},[649,460],[17,4683,4685,4688],{"className":4684},[61,460],[17,4686,3928],{"className":4687,"style":4138},[61,62,460],[17,4689,4691],{"className":4690},[433],[17,4692,4694,4718],{"className":4693},[437,1313],[17,4695,4697,4715],{"className":4696},[441],[17,4698,4700],{"className":4699,"style":4541},[445],[17,4701,4703,4706],{"style":4702},"top:-2.357em;margin-left:-0.0572em;margin-right:0.0714em;",[17,4704],{"className":4705,"style":4548},[452],[17,4707,4709],{"className":4708},[457,4552,4553,460],[17,4710,4712],{"className":4711},[61,460],[17,4713,3544],{"className":4714},[61,460],[17,4716,1346],{"className":4717},[1345],[17,4719,4721],{"className":4720},[441],[17,4722,4724],{"className":4723,"style":4569},[445],[17,4725],{},[17,4727,3557],{"className":4728},[233,460],[17,4730,4732,4735],{"className":4731},[61,460],[17,4733,3928],{"className":4734,"style":4138},[61,62,460],[17,4736,4738],{"className":4737},[433],[17,4739,4741,4764],{"className":4740},[437,1313],[17,4742,4744,4761],{"className":4743},[441],[17,4745,4747],{"className":4746,"style":4541},[445],[17,4748,4749,4752],{"style":4702},[17,4750],{"className":4751,"style":4548},[452],[17,4753,4755],{"className":4754},[457,4552,4553,460],[17,4756,4758],{"className":4757},[61,460],[17,4759,410],{"className":4760},[61,460],[17,4762,1346],{"className":4763},[1345],[17,4765,4767],{"className":4766},[441],[17,4768,4770],{"className":4769,"style":4569},[445],[17,4771],{},[17,4773,3557],{"className":4774},[233,460],[17,4776,3905],{"className":4777},[1301,460],[17,4779,3557],{"className":4780},[233,460],[17,4782,4784,4787],{"className":4783},[61,460],[17,4785,3928],{"className":4786,"style":4138},[61,62,460],[17,4788,4790],{"className":4789},[433],[17,4791,4793,4817],{"className":4792},[437,1313],[17,4794,4796,4814],{"className":4795},[441],[17,4797,4799],{"className":4798,"style":4645},[445],[17,4800,4802,4805],{"style":4801},"top:-2.3488em;margin-left:-0.0572em;margin-right:0.0714em;",[17,4803],{"className":4804,"style":4548},[452],[17,4806,4808],{"className":4807},[457,4552,4553,460],[17,4809,4811],{"className":4810},[61,460],[17,4812,39],{"className":4813,"style":63},[61,62,460],[17,4815,1346],{"className":4816},[1345],[17,4818,4820],{"className":4819},[441],[17,4821,4823],{"className":4822,"style":4670},[445],[17,4824],{},[17,4826,632],{"className":4827},[674,460]," to form the algebraic cofactor of ",[17,4830,4832,4845],{"className":4831,"translate":21},[20],[17,4833,4835],{"className":4834},[25],[27,4836,4837],{"xmlns":29},[31,4838,4839,4843],{},[34,4840,4841],{},[37,4842,172],{},[41,4844,172],{"encoding":43},[17,4846,4848],{"className":4847,"ariaHidden":48},[47],[17,4849,4851,4854],{"className":4850},[52],[17,4852],{"className":4853,"style":123},[56],[17,4855,172],{"className":4856,"style":187},[61,62],[10,4858,4859,4862,4863,4891,4892,4920,4921,4950,4951,4979],{},[13,4860,4861],{},"Lemma:"," Every term in the product of any minor ",[17,4864,4866,4879],{"className":4865,"translate":21},[20],[17,4867,4869],{"className":4868},[25],[27,4870,4871],{"xmlns":29},[31,4872,4873,4877],{},[34,4874,4875],{},[37,4876,172],{},[41,4878,172],{"encoding":43},[17,4880,4882],{"className":4881,"ariaHidden":48},[47],[17,4883,4885,4888],{"className":4884},[52],[17,4886],{"className":4887,"style":123},[56],[17,4889,172],{"className":4890,"style":187},[61,62]," of determinant ",[17,4893,4895,4908],{"className":4894,"translate":21},[20],[17,4896,4898],{"className":4897},[25],[27,4899,4900],{"xmlns":29},[31,4901,4902,4906],{},[34,4903,4904],{},[37,4905,111],{},[41,4907,111],{"encoding":43},[17,4909,4911],{"className":4910,"ariaHidden":48},[47],[17,4912,4914,4917],{"className":4913},[52],[17,4915],{"className":4916,"style":123},[56],[17,4918,111],{"className":4919,"style":127},[61,62]," and its algebraic cofactor ",[17,4922,4924,4938],{"className":4923,"translate":21},[20],[17,4925,4927],{"className":4926},[25],[27,4928,4929],{"xmlns":29},[31,4930,4931,4936],{},[34,4932,4933],{},[37,4934,4935],{},"A",[41,4937,4935],{"encoding":43},[17,4939,4941],{"className":4940,"ariaHidden":48},[47],[17,4942,4944,4947],{"className":4943},[52],[17,4945],{"className":4946,"style":123},[56],[17,4948,4935],{"className":4949},[61,62]," corresponds to a term in the expansion of determinant ",[17,4952,4954,4967],{"className":4953,"translate":21},[20],[17,4955,4957],{"className":4956},[25],[27,4958,4959],{"xmlns":29},[31,4960,4961,4965],{},[34,4962,4963],{},[37,4964,111],{},[41,4966,111],{"encoding":43},[17,4968,4970],{"className":4969,"ariaHidden":48},[47],[17,4971,4973,4976],{"className":4972},[52],[17,4974],{"className":4975,"style":123},[56],[17,4977,111],{"className":4978,"style":127},[61,62],", with the same sign.",[10,4981,4982,4985,4986,5014,5015,5043],{},[13,4983,4984],{},"Proof:"," First, consider the case where ",[17,4987,4989,5002],{"className":4988,"translate":21},[20],[17,4990,4992],{"className":4991},[25],[27,4993,4994],{"xmlns":29},[31,4995,4996,5000],{},[34,4997,4998],{},[37,4999,172],{},[41,5001,172],{"encoding":43},[17,5003,5005],{"className":5004,"ariaHidden":48},[47],[17,5006,5008,5011],{"className":5007},[52],[17,5009],{"className":5010,"style":123},[56],[17,5012,172],{"className":5013,"style":187},[61,62]," is located in the upper-left corner of 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this case, the algebraic cofactor of ",[17,7664,7666,7679],{"className":7665,"translate":21},[20],[17,7667,7669],{"className":7668},[25],[27,7670,7671],{"xmlns":29},[31,7672,7673,7677],{},[34,7674,7675],{},[37,7676,172],{},[41,7678,172],{"encoding":43},[17,7680,7682],{"className":7681,"ariaHidden":48},[47],[17,7683,7685,7688],{"className":7684},[52],[17,7686],{"className":7687,"style":123},[56],[17,7689,172],{"className":7690,"style":187},[61,62]," is:",[17,7693,7695],{"className":7694,"translate":21},[807],[17,7696,7698,7778],{"className":7697,"translate":21},[20],[17,7699,7701],{"className":7700},[25],[27,7702,7703],{"xmlns":29,"display":816},[31,7704,7705,7775],{},[34,7706,7707,7709,7711,7713,7715,7717,7761,7767,7769],{},[37,7708,4935],{},[204,7710,825],{},[204,7712,622],{"stretchy":621},[204,7714,627],{},[408,7716,3544],{},[403,7718,7719,7721],{},[204,7720,632],{"stretchy":621},[34,7722,7723,7725,7727,7729,7731,7733,7735,7737,7739,7741,7743,7745,7747,7749,7751,7753,7755,7757,7759],{},[204,7724,622],{"stretchy":621},[408,7726,3544],{},[204,7728,3557],{},[408,7730,410],{},[204,7732,3557],{},[204,7734,3905],{},[204,7736,3557],{},[37,7738,39],{},[204,7740,632],{"stretchy":621},[204,7742,3557],{},[204,7744,622],{"stretchy":621},[408,7746,3544],{},[204,7748,3557],{},[408,7750,410],{},[204,7752,3557],{},[204,7754,3905],{},[204,7756,3557],{},[37,7758,39],{},[204,7760,632],{"stretchy":621},[403,7762,7763,7765],{},[37,7764,172],{},[204,7766,695],{"mathvariant":693,"lspace":694,"rspace":694},[204,7768,825],{},[403,7770,7771,7773],{},[37,7772,172],{},[204,7774,695],{"mathvariant":693,"lspace":694,"rspace":694},[41,7776,7777],{"encoding":43},"A=(-1)^{(1+2+\\cdots +k)+(1+2+\\cdots+k)}M'=M'",[17,7779,7781,7799,7941],{"className":7780,"ariaHidden":48},[47],[17,7782,7784,7787,7790,7793,7796],{"className":7783},[52],[17,7785],{"className":7786,"style":123},[56],[17,7788,4935],{"className":7789},[61,62],[17,7791],{"className":7792,"style":372},[228],[17,7794,825],{"className":7795},[376],[17,7797],{"className":7798,"style":372},[228],[17,7800,7802,7805,7808,7811,7814,7900,7932,7935,7938],{"className":7801},[52],[17,7803],{"className":7804,"style":3613},[56],[17,7806,622],{"className":7807},[649],[17,7809,627],{"className":7810},[61],[17,7812,3544],{"className":7813},[61],[17,7815,7817,7820],{"className":7816},[674],[17,7818,632],{"className":7819},[674],[17,7821,7823],{"className":7822},[433],[17,7824,7826],{"className":7825},[437],[17,7827,7829],{"className":7828},[441],[17,7830,7832],{"className":7831,"style":3641},[445],[17,7833,7834,7837],{"style":3110},[17,7835],{"className":7836,"style":453},[452],[17,7838,7840],{"className":7839},[457,458,459,460],[17,7841,7843,7846,7849,7852,7855,7858,7861,7864,7867,7870,7873,7876,7879,7882,7885,7888,7891,7894,7897],{"className":7842},[61,460],[17,7844,622],{"className":7845},[649,460],[17,7847,3544],{"className":7848},[61,460],[17,7850,3557],{"className":7851},[233,460],[17,7853,410],{"className":7854},[61,460],[17,7856,3557],{"className":7857},[233,460],[17,7859,3905],{"className":7860},[1301,460],[17,7862,3557],{"className":7863},[233,460],[17,7865,39],{"className":7866,"style":63},[61,62,460],[17,7868,632],{"className":7869},[674,460],[17,7871,3557],{"className":7872},[233,460],[17,7874,622],{"className":7875},[649,460],[17,7877,3544],{"className":7878},[61,460],[17,7880,3557],{"className":7881},[233,460],[17,7883,410],{"className":7884},[61,460],[17,7886,3557],{"className":7887},[233,460],[17,7889,3905],{"className":7890},[1301,460],[17,7892,3557],{"className":7893},[233,460],[17,7895,39],{"className":7896,"style":63},[61,62,460],[17,7898,632],{"className":7899},[674,460],[17,7901,7903,7906],{"className":7902},[61],[17,7904,172],{"className":7905,"style":187},[61,62],[17,7907,7909],{"className":7908},[433],[17,7910,7912],{"className":7911},[437],[17,7913,7915],{"className":7914},[441],[17,7916,7918],{"className":7917,"style":3089},[445],[17,7919,7920,7923],{"style":3110},[17,7921],{"className":7922,"style":453},[452],[17,7924,7926],{"className":7925},[457,458,459,460],[17,7927,7929],{"className":7928},[61,460],[17,7930,695],{"className":7931},[61,460],[17,7933],{"className":7934,"style":372},[228],[17,7936,825],{"className":7937},[376],[17,7939],{"className":7940,"style":372},[228],[17,7942,7944,7947],{"className":7943},[52],[17,7945],{"className":7946,"style":3089},[56],[17,7948,7950,7953],{"className":7949},[61],[17,7951,172],{"className":7952,"style":187},[61,62],[17,7954,7956],{"className":7955},[433],[17,7957,7959],{"className":7958},[437],[17,7960,7962],{"className":7961},[441],[17,7963,7965],{"className":7964,"style":3089},[445],[17,7966,7967,7970],{"style":3110},[17,7968],{"className":7969,"style":453},[452],[17,7971,7973],{"className":7972},[457,458,459,460],[17,7974,7976],{"className":7975},[61,460],[17,7977,695],{"className":7978},[61,460],[10,7980,7981,7982,8010,8011,8353,8354,8538,8539,8616,8617,798],{},"Each term in ",[17,7983,7985,7998],{"className":7984,"translate":21},[20],[17,7986,7988],{"className":7987},[25],[27,7989,7990],{"xmlns":29},[31,7991,7992,7996],{},[34,7993,7994],{},[37,7995,172],{},[41,7997,172],{"encoding":43},[17,7999,8001],{"className":8000,"ariaHidden":48},[47],[17,8002,8004,8007],{"className":8003},[52],[17,8005],{"className":8006,"style":123},[56],[17,8008,172],{"className":8009,"style":187},[61,62]," can be written as ",[17,8012,8014,8071],{"className":8013,"translate":21},[20],[17,8015,8017],{"className":8016},[25],[27,8018,8019],{"xmlns":29},[31,8020,8021,8068],{},[34,8022,8023,8038,8052,8054],{},[848,8024,8025,8027],{},[37,8026,852],{},[34,8028,8029,8031],{},[408,8030,3544],{},[848,8032,8033,8036],{},[37,8034,8035],{},"α",[408,8037,3544],{},[848,8039,8040,8042],{},[37,8041,852],{},[34,8043,8044,8046],{},[408,8045,410],{},[848,8047,8048,8050],{},[37,8049,8035],{},[408,8051,410],{},[204,8053,3905],{},[848,8055,8056,8058],{},[37,8057,852],{},[34,8059,8060,8062],{},[37,8061,39],{},[848,8063,8064,8066],{},[37,8065,8035],{},[37,8067,39],{},[41,8069,8070],{"encoding":43},"a_{1\\alpha_{1}}a_{2\\alpha_{2}}\\cdots a_{k\\alpha_{k}}",[17,8072,8074],{"className":8073,"ariaHidden":48},[47],[17,8075,8077,8081,8170,8256,8259,8262,8265],{"className":8076},[52],[17,8078],{"className":8079,"style":8080},[56],"height:0.6864em;vertical-align:-0.2559em;",[17,8082,8084,8087],{"className":8083},[61],[17,8085,852],{"className":8086},[61,62],[17,8088,8090],{"className":8089},[433],[17,8091,8093,8161],{"className":8092},[437,1313],[17,8094,8096,8158],{"className":8095},[441],[17,8097,8099],{"className":8098,"style":1404},[445],[17,8100,8101,8104],{"style":1407},[17,8102],{"className":8103,"style":453},[452],[17,8105,8107],{"className":8106},[457,458,459,460],[17,8108,8110,8113],{"className":8109},[61,460],[17,8111,3544],{"className":8112},[61,460],[17,8114,8116,8120],{"className":8115},[61,460],[17,8117,8035],{"className":8118,"style":8119},[61,62,460],"margin-right:0.0037em;",[17,8121,8123],{"className":8122},[433],[17,8124,8126,8150],{"className":8125},[437,1313],[17,8127,8129,8147],{"className":8128},[441],[17,8130,8132],{"className":8131,"style":4541},[445],[17,8133,8135,8138],{"style":8134},"top:-2.357em;margin-left:-0.0037em;margin-right:0.0714em;",[17,8136],{"className":8137,"style":4548},[452],[17,8139,8141],{"className":8140},[457,4552,4553,460],[17,8142,8144],{"className":8143},[61,460],[17,8145,3544],{"className":8146},[61,460],[17,8148,1346],{"className":8149},[1345],[17,8151,8153],{"className":8152},[441],[17,8154,8156],{"className":8155,"style":4569},[445],[17,8157],{},[17,8159,1346],{"className":8160},[1345],[17,8162,8164],{"className":8163},[441],[17,8165,8168],{"className":8166,"style":8167},[445],"height:0.2501em;",[17,8169],{},[17,8171,8173,8176],{"className":8172},[61],[17,8174,852],{"className":8175},[61,62],[17,8177,8179],{"className":8178},[433],[17,8180,8182,8248],{"className":8181},[437,1313],[17,8183,8185,8245],{"className":8184},[441],[17,8186,8188],{"className":8187,"style":1404},[445],[17,8189,8190,8193],{"style":1407},[17,8191],{"className":8192,"style":453},[452],[17,8194,8196],{"className":8195},[457,458,459,460],[17,8197,8199,8202],{"className":8198},[61,460],[17,8200,410],{"className":8201},[61,460],[17,8203,8205,8208],{"className":8204},[61,460],[17,8206,8035],{"className":8207,"style":8119},[61,62,460],[17,8209,8211],{"className":8210},[433],[17,8212,8214,8237],{"className":8213},[437,1313],[17,8215,8217,8234],{"className":8216},[441],[17,8218,8220],{"className":8219,"style":4541},[445],[17,8221,8222,8225],{"style":8134},[17,8223],{"className":8224,"style":4548},[452],[17,8226,8228],{"className":8227},[457,4552,4553,460],[17,8229,8231],{"className":8230},[61,460],[17,8232,410],{"className":8233},[61,460],[17,8235,1346],{"className":8236},[1345],[17,8238,8240],{"className":8239},[441],[17,8241,8243],{"className":8242,"style":4569},[445],[17,8244],{},[17,8246,1346],{"className":8247},[1345],[17,8249,8251],{"className":8250},[441],[17,8252,8254],{"className":8253,"style":8167},[445],[17,8255],{},[17,8257],{"className":8258,"style":2406},[228],[17,8260,3905],{"className":8261},[1301],[17,8263],{"className":8264,"style":2406},[228],[17,8266,8268,8271],{"className":8267},[61],[17,8269,852],{"className":8270},[61,62],[17,8272,8274],{"className":8273},[433],[17,8275,8277,8344],{"className":8276},[437,1313],[17,8278,8280,8341],{"className":8279},[441],[17,8281,8283],{"className":8282,"style":4097},[445],[17,8284,8285,8288],{"style":1407},[17,8286],{"className":8287,"style":453},[452],[17,8289,8291],{"className":8290},[457,458,459,460],[17,8292,8294,8297],{"className":8293},[61,460],[17,8295,39],{"className":8296,"style":63},[61,62,460],[17,8298,8300,8303],{"className":8299},[61,460],[17,8301,8035],{"className":8302,"style":8119},[61,62,460],[17,8304,8306],{"className":8305},[433],[17,8307,8309,8333],{"className":8308},[437,1313],[17,8310,8312,8330],{"className":8311},[441],[17,8313,8315],{"className":8314,"style":4645},[445],[17,8316,8318,8321],{"style":8317},"top:-2.3488em;margin-left:-0.0037em;margin-right:0.0714em;",[17,8319],{"className":8320,"style":4548},[452],[17,8322,8324],{"className":8323},[457,4552,4553,460],[17,8325,8327],{"className":8326},[61,460],[17,8328,39],{"className":8329,"style":63},[61,62,460],[17,8331,1346],{"className":8332},[1345],[17,8334,8336],{"className":8335},[441],[17,8337,8339],{"className":8338,"style":4670},[445],[17,8340],{},[17,8342,1346],{"className":8343},[1345],[17,8345,8347],{"className":8346},[441],[17,8348,8351],{"className":8349,"style":8350},[445],"height:0.2559em;",[17,8352],{},", where ",[17,8355,8357,8389],{"className":8356,"translate":21},[20],[17,8358,8360],{"className":8359},[25],[27,8361,8362],{"xmlns":29},[31,8363,8364,8386],{},[34,8365,8366,8372,8378,8380],{},[848,8367,8368,8370],{},[37,8369,8035],{},[408,8371,3544],{},[848,8373,8374,8376],{},[37,8375,8035],{},[408,8377,410],{},[204,8379,3905],{},[848,8381,8382,8384],{},[37,8383,8035],{},[37,8385,39],{},[41,8387,8388],{"encoding":43},"\\alpha_{1}\\alpha_{2}\\cdots\\alpha_{k}",[17,8390,8392],{"className":8391,"ariaHidden":48},[47],[17,8393,8395,8399,8443,8486,8489,8492,8495],{"className":8394},[52],[17,8396],{"className":8397,"style":8398},[56],"height:0.5806em;vertical-align:-0.15em;",[17,8400,8402,8405],{"className":8401},[61],[17,8403,8035],{"className":8404,"style":8119},[61,62],[17,8406,8408],{"className":8407},[433],[17,8409,8411,8435],{"className":8410},[437,1313],[17,8412,8414,8432],{"className":8413},[441],[17,8415,8417],{"className":8416,"style":1404},[445],[17,8418,8420,8423],{"style":8419},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[17,8421],{"className":8422,"style":453},[452],[17,8424,8426],{"className":8425},[457,458,459,460],[17,8427,8429],{"className":8428},[61,460],[17,8430,3544],{"className":8431},[61,460],[17,8433,1346],{"className":8434},[1345],[17,8436,8438],{"className":8437},[441],[17,8439,8441],{"className":8440,"style":1429},[445],[17,8442],{},[17,8444,8446,8449],{"className":8445},[61],[17,8447,8035],{"className":8448,"style":8119},[61,62],[17,8450,8452],{"className":8451},[433],[17,8453,8455,8478],{"className":8454},[437,1313],[17,8456,8458,8475],{"className":8457},[441],[17,8459,8461],{"className":8460,"style":1404},[445],[17,8462,8463,8466],{"style":8419},[17,8464],{"className":8465,"style":453},[452],[17,8467,8469],{"className":8468},[457,458,459,460],[17,8470,8472],{"className":8471},[61,460],[17,8473,410],{"className":8474},[61,460],[17,8476,1346],{"className":8477},[1345],[17,8479,8481],{"className":8480},[441],[17,8482,8484],{"className":8483,"style":1429},[445],[17,8485],{},[17,8487],{"className":8488,"style":2406},[228],[17,8490,3905],{"className":8491},[1301],[17,8493],{"className":8494,"style":2406},[228],[17,8496,8498,8501],{"className":8497},[61],[17,8499,8035],{"className":8500,"style":8119},[61,62],[17,8502,8504],{"className":8503},[433],[17,8505,8507,8530],{"className":8506},[437,1313],[17,8508,8510,8527],{"className":8509},[441],[17,8511,8513],{"className":8512,"style":4097},[445],[17,8514,8515,8518],{"style":8419},[17,8516],{"className":8517,"style":453},[452],[17,8519,8521],{"className":8520},[457,458,459,460],[17,8522,8524],{"className":8523},[61,460],[17,8525,39],{"className":8526,"style":63},[61,62,460],[17,8528,1346],{"className":8529},[1345],[17,8531,8533],{"className":8532},[441],[17,8534,8536],{"className":8535,"style":1429},[445],[17,8537],{}," is a permutation of 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product of these two terms is 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a_{k\\alpha_{k}}a_{k+1,\\beta_{k+1}}\\cdots 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and the combined sign 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(-1)^{\\tau(\\alpha_{1}\\alpha_{2}\\cdots \\alpha_{k}\\beta_{k+1}\\cdots \\beta_{n}) 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7,11320],{"className":11313},[61,460],[17,11315,39],{"className":11316,"style":63},[61,62,460],[17,11318,3557],{"className":11319},[233,460],[17,11321,3544],{"className":11322},[61,460],[17,11324,1346],{"className":11325},[1345],[17,11327,11329],{"className":11328},[441],[17,11330,11332],{"className":11331,"style":9098},[445],[17,11333],{},[17,11335,3905],{"className":11336},[1301,460],[17,11338,11340,11343],{"className":11339},[61,460],[17,11341,8944],{"className":11342,"style":9055},[61,62,460],[17,11344,11346],{"className":11345},[433],[17,11347,11349,11372],{"className":11348},[437,1313],[17,11350,11352,11369],{"className":11351},[441],[17,11353,11355],{"className":11354,"style":9273},[445],[17,11356,11357,11360],{"style":9276},[17,11358],{"className":11359,"style":4548},[452],[17,11361,11363],{"className":11362},[457,4552,4553,460],[17,11364,11366],{"className":11365},[61,460],[17,11367,80],{"className":11368},[61,62,460],[17,11370,1346],{"className":11371},[1345],[17,11373,11375],{"className":11374},[441],[17,11376,11378],{"className":11377,"style":4569},[445],[17,11379],{},[17,11381,632],{"className":11382},[674,460],[10,11384,11385,11386,11414],{},"Thus, this product is a term in the expansion of determinant ",[17,11387,11389,11402],{"className":11388,"translate":21},[20],[17,11390,11392],{"className":11391},[25],[27,11393,11394],{"xmlns":29},[31,11395,11396,11400],{},[34,11397,11398],{},[37,11399,111],{},[41,11401,111],{"encoding":43},[17,11403,11405],{"className":11404,"ariaHidden":48},[47],[17,11406,11408,11411],{"className":11407},[52],[17,11409],{"className":11410,"style":123},[56],[17,11412,111],{"className":11413,"style":127},[61,62]," with the same sign.",[10,11416,11417],{},"Now, we will prove the general case.",[10,11419,11420,11421,11449,11450,11649,11650,4352,11849,8353,11877,12097,12098,798],{},"Let minor ",[17,11422,11424,11437],{"className":11423,"translate":21},[20],[17,11425,11427],{"className":11426},[25],[27,11428,11429],{"xmlns":29},[31,11430,11431,11435],{},[34,11432,11433],{},[37,11434,172],{},[41,11436,172],{"encoding":43},[17,11438,11440],{"className":11439,"ariaHidden":48},[47],[17,11441,11443,11446],{"className":11442},[52],[17,11444],{"className":11445,"style":123},[56],[17,11447,172],{"className":11448,"style":187},[61,62]," be located in rows ",[17,11451,11453,11493],{"className":11452,"translate":21},[20],[17,11454,11456],{"className":11455},[25],[27,11457,11458],{"xmlns":29},[31,11459,11460,11490],{},[34,11461,11462,11468,11470,11476,11478,11480,11482,11484],{},[848,11463,11464,11466],{},[37,11465,3890],{},[408,11467,3544],{},[204,11469,1084],{"separator":48},[848,11471,11472,11474],{},[37,11473,3890],{},[408,11475,410],{},[204,11477,1084],{"separator":48},[204,11479,3905],{},[3907,11481,3909],{},[204,11483,1084],{"separator":48},[848,11485,11486,11488],{},[37,11487,3890],{},[37,11489,39],{},[41,11491,11492],{"encoding":43},"i_1, i_2, \\cdots, i_k",[17,11494,11496],{"className":11495,"ariaHidden":48},[47],[17,11497,11499,11502,11542,11545,11548,11588,11591,11594,11597,11600,11603,11606,11609],{"className":11498},[52],[17,11500],{"className":11501,"style":3965},[56],[17,11503,11505,11508],{"className":11504},[61],[17,11506,3890],{"className":11507},[61,62],[17,11509,11511],{"className":11510},[433],[17,11512,11514,11534],{"className":11513},[437,1313],[17,11515,11517,11531],{"className":11516},[441],[17,11518,11520],{"className":11519,"style":1404},[445],[17,11521,11522,11525],{"style":1407},[17,11523],{"className":11524,"style":453},[452],[17,11526,11528],{"className":11527},[457,458,459,460],[17,11529,3544],{"className":11530},[61,460],[17,11532,1346],{"className":11533},[1345],[17,11535,11537],{"className":11536},[441],[17,11538,11540],{"className":11539,"style":1429},[445],[17,11541],{},[17,11543,1084],{"className":11544},[2410],[17,11546],{"className":11547,"style":2406},[228],[17,11549,11551,11554],{"className":11550},[61],[17,11552,3890],{"className":11553},[61,62],[17,11555,11557],{"className":11556},[433],[17,11558,11560,11580],{"className":11559},[437,1313],[17,11561,11563,11577],{"className":11562},[441],[17,11564,11566],{"className":11565,"style":1404},[445],[17,11567,11568,11571],{"style":1407},[17,11569],{"className":11570,"style":453},[452],[17,11572,11574],{"className":11573},[457,458,459,460],[17,11575,410],{"className":11576},[61,460],[17,11578,1346],{"className":11579},[1345],[17,11581,11583],{"className":11582},[441],[17,11584,11586],{"className":11585,"style":1429},[445],[17,11587],{},[17,11589,1084],{"className":11590},[2410],[17,11592],{"className":11593,"style":2406},[228],[17,11595,3905],{"className":11596},[1301],[17,11598],{"className":11599,"style":2406},[228],[17,11601],{"className":11602,"style":2406},[228],[17,11604,1084],{"className":11605},[2410],[17,11607],{"className":11608,"style":2406},[228],[17,11610,11612,11615],{"className":11611},[61],[17,11613,3890],{"className":11614},[61,62],[17,11616,11618],{"className":11617},[433],[17,11619,11621,11641],{"className":11620},[437,1313],[17,11622,11624,11638],{"className":11623},[441],[17,11625,11627],{"className":11626,"style":4097},[445],[17,11628,11629,11632],{"style":1407},[17,11630],{"className":11631,"style":453},[452],[17,11633,11635],{"className":11634},[457,458,459,460],[17,11636,39],{"className":11637,"style":63},[61,62,460],[17,11639,1346],{"className":11640},[1345],[17,11642,11644],{"className":11643},[441],[17,11645,11647],{"className":11646,"style":1429},[445],[17,11648],{}," and columns ",[17,11651,11653,11693],{"className":11652,"translate":21},[20],[17,11654,11656],{"className":11655},[25],[27,11657,11658],{"xmlns":29},[31,11659,11660,11690],{},[34,11661,11662,11668,11670,11676,11678,11680,11682,11684],{},[848,11663,11664,11666],{},[37,11665,3928],{},[408,11667,3544],{},[204,11669,1084],{"separator":48},[848,11671,11672,11674],{},[37,11673,3928],{},[408,11675,410],{},[204,11677,1084],{"separator":48},[204,11679,3905],{},[3907,11681,3909],{},[204,11683,1084],{"separator":48},[848,11685,11686,11688],{},[37,11687,3928],{},[37,11689,39],{},[41,11691,11692],{"encoding":43},"j_1, j_2, \\cdots, j_k",[17,11694,11696],{"className":11695,"ariaHidden":48},[47],[17,11697,11699,11702,11742,11745,11748,11788,11791,11794,11797,11800,11803,11806,11809],{"className":11698},[52],[17,11700],{"className":11701,"style":3965},[56],[17,11703,11705,11708],{"className":11704},[61],[17,11706,3928],{"className":11707,"style":4138},[61,62],[17,11709,11711],{"className":11710},[433],[17,11712,11714,11734],{"className":11713},[437,1313],[17,11715,11717,11731],{"className":11716},[441],[17,11718,11720],{"className":11719,"style":1404},[445],[17,11721,11722,11725],{"style":4153},[17,11723],{"className":11724,"style":453},[452],[17,11726,11728],{"className":11727},[457,458,459,460],[17,11729,3544],{"className":11730},[61,460],[17,11732,1346],{"className":11733},[1345],[17,11735,11737],{"className":11736},[441],[17,11738,11740],{"className":11739,"style":1429},[445],[17,11741],{},[17,11743,1084],{"className":11744},[2410],[17,11746],{"className":11747,"style":2406},[228],[17,11749,11751,11754],{"className":11750},[61],[17,11752,3928],{"className":11753,"style":4138},[61,62],[17,11755,11757],{"className":11756},[433],[17,11758,11760,11780],{"className":11759},[437,1313],[17,11761,11763,11777],{"className":11762},[441],[17,11764,11766],{"className":11765,"style":1404},[445],[17,11767,11768,11771],{"style":4153},[17,11769],{"className":11770,"style":453},[452],[17,11772,11774],{"className":11773},[457,458,459,460],[17,11775,410],{"className":11776},[61,460],[17,11778,1346],{"className":11779},[1345],[17,11781,11783],{"className":11782},[441],[17,11784,11786],{"className":11785,"style":1429},[445],[17,11787],{},[17,11789,1084],{"className":11790},[2410],[17,11792],{"className":11793,"style":2406},[228],[17,11795,3905],{"className":11796},[1301],[17,11798],{"className":11799,"style":2406},[228],[17,11801],{"className":11802,"style":2406},[228],[17,11804,1084],{"className":11805},[2410],[17,11807],{"className":11808,"style":2406},[228],[17,11810,11812,11815],{"className":11811},[61],[17,11813,3928],{"className":11814,"style":4138},[61,62],[17,11816,11818],{"className":11817},[433],[17,11819,11821,11841],{"className":11820},[437,1313],[17,11822,11824,11838],{"className":11823},[441],[17,11825,11827],{"className":11826,"style":4097},[445],[17,11828,11829,11832],{"style":4153},[17,11830],{"className":11831,"style":453},[452],[17,11833,11835],{"className":11834},[457,458,459,460],[17,11836,39],{"className":11837,"style":63},[61,62,460],[17,11839,1346],{"className":11840},[1345],[17,11842,11844],{"className":11843},[441],[17,11845,11847],{"className":11846,"style":1429},[445],[17,11848],{},[17,11850,11852,11865],{"className":11851,"translate":21},[20],[17,11853,11855],{"className":11854},[25],[27,11856,11857],{"xmlns":29},[31,11858,11859,11863],{},[34,11860,11861],{},[37,11862,111],{},[41,11864,111],{"encoding":43},[17,11866,11868],{"className":11867,"ariaHidden":48},[47],[17,11869,11871,11874],{"className":11870},[52],[17,11872],{"className":11873,"style":123},[56],[17,11875,111],{"className":11876,"style":127},[61,62],[17,11878,11880,11918],{"className":11879,"translate":21},[20],[17,11881,11883],{"className":11882},[25],[27,11884,11885],{"xmlns":29},[31,11886,11887,11915],{},[34,11888,11889,11895,11897,11903,11905,11907,11909],{},[848,11890,11891,11893],{},[37,11892,3890],{},[408,11894,3544],{},[204,11896,484],{},[848,11898,11899,11901],{},[37,11900,3890],{},[408,11902,410],{},[204,11904,484],{},[204,11906,3905],{},[204,11908,484],{},[848,11910,11911,11913],{},[37,11912,3890],{},[37,11914,39],{},[41,11916,11917],{"encoding":43},"i_1 \u003C i_2 \u003C \\cdots \u003C i_k",[17,11919,11921,11977,12032,12051],{"className":11920,"ariaHidden":48},[47],[17,11922,11924,11928,11968,11971,11974],{"className":11923},[52],[17,11925],{"className":11926,"style":11927},[56],"height:0.8095em;vertical-align:-0.15em;",[17,11929,11931,11934],{"className":11930},[61],[17,11932,3890],{"className":11933},[61,62],[17,11935,11937],{"className":11936},[433],[17,11938,11940,11960],{"className":11939},[437,1313],[17,11941,11943,11957],{"className":11942},[441],[17,11944,11946],{"className":11945,"style":1404},[445],[17,11947,11948,11951],{"style":1407},[17,11949],{"className":11950,"style":453},[452],[17,11952,11954],{"className":11953},[457,458,459,460],[17,11955,3544],{"className":11956},[61,460],[17,11958,1346],{"className":11959},[1345],[17,11961,11963],{"className":11962},[441],[17,11964,11966],{"className":11965,"style":1429},[445],[17,11967],{},[17,11969],{"className":11970,"style":372},[228],[17,11972,484],{"className":11973},[376],[17,11975],{"className":11976,"style":372},[228],[17,11978,11980,11983,12023,12026,12029],{"className":11979},[52],[17,11981],{"className":11982,"style":11927},[56],[17,11984,11986,11989],{"className":11985},[61],[17,11987,3890],{"className":11988},[61,62],[17,11990,11992],{"className":11991},[433],[17,11993,11995,12015],{"className":11994},[437,1313],[17,11996,11998,12012],{"className":11997},[441],[17,11999,12001],{"className":12000,"style":1404},[445],[17,12002,12003,12006],{"style":1407},[17,12004],{"className":12005,"style":453},[452],[17,12007,12009],{"className":12008},[457,458,459,460],[17,12010,410],{"className":12011},[61,460],[17,12013,1346],{"className":12014},[1345],[17,12016,12018],{"className":12017},[441],[17,12019,12021],{"className":12020,"style":1429},[445],[17,12022],{},[17,12024],{"className":12025,"style":372},[228],[17,12027,484],{"className":12028},[376],[17,12030],{"className":12031,"style":372},[228],[17,12033,12035,12039,12042,12045,12048],{"className":12034},[52],[17,12036],{"className":12037,"style":12038},[56],"height:0.5782em;vertical-align:-0.0391em;",[17,12040,3905],{"className":12041},[1301],[17,12043],{"className":12044,"style":372},[228],[17,12046,484],{"className":12047},[376],[17,12049],{"className":12050,"style":372},[228],[17,12052,12054,12057],{"className":12053},[52],[17,12055],{"className":12056,"style":11927},[56],[17,12058,12060,12063],{"className":12059},[61],[17,12061,3890],{"className":12062},[61,62],[17,12064,12066],{"className":12065},[433],[17,12067,12069,12089],{"className":12068},[437,1313],[17,12070,12072,12086],{"className":12071},[441],[17,12073,12075],{"className":12074,"style":4097},[445],[17,12076,12077,12080],{"style":1407},[17,12078],{"className":12079,"style":453},[452],[17,12081,12083],{"className":12082},[457,458,459,460],[17,12084,39],{"className":12085,"style":63},[61,62,460],[17,12087,1346],{"className":12088},[1345],[17,12090,12092],{"className":12091},[441],[17,12093,12095],{"className":12094,"style":1429},[445],[17,12096],{},", and ",[17,12099,12101,12139],{"className":12100,"translate":21},[20],[17,12102,12104],{"className":12103},[25],[27,12105,12106],{"xmlns":29},[31,12107,12108,12136],{},[34,12109,12110,12116,12118,12124,12126,12128,12130],{},[848,12111,12112,12114],{},[37,12113,3928],{},[408,12115,3544],{},[204,12117,484],{},[848,12119,12120,12122],{},[37,12121,3928],{},[408,12123,410],{},[204,12125,484],{},[204,12127,3905],{},[204,12129,484],{},[848,12131,12132,12134],{},[37,12133,3928],{},[37,12135,39],{},[41,12137,12138],{"encoding":43},"j_1 \u003C j_2 \u003C \\cdots \u003C j_k",[17,12140,12142,12197,12252,12270],{"className":12141,"ariaHidden":48},[47],[17,12143,12145,12148,12188,12191,12194],{"className":12144},[52],[17,12146],{"className":12147,"style":3965},[56],[17,12149,12151,12154],{"className":12150},[61],[17,12152,3928],{"className":12153,"style":4138},[61,62],[17,12155,12157],{"className":12156},[433],[17,12158,12160,12180],{"className":12159},[437,1313],[17,12161,12163,12177],{"className":12162},[441],[17,12164,12166],{"className":12165,"style":1404},[445],[17,12167,12168,12171],{"style":4153},[17,12169],{"className":12170,"style":453},[452],[17,12172,12174],{"className":12173},[457,458,459,460],[17,12175,3544],{"className":12176},[61,460],[17,12178,1346],{"className":12179},[1345],[17,12181,12183],{"className":12182},[441],[17,12184,12186],{"className":12185,"style":1429},[445],[17,12187],{},[17,12189],{"className":12190,"style":372},[228],[17,12192,484],{"className":12193},[376],[17,12195],{"className":12196,"style":372},[228],[17,12198,12200,12203,12243,12246,12249],{"className":12199},[52],[17,12201],{"className":12202,"style":3965},[56],[17,12204,12206,12209],{"className":12205},[61],[17,12207,3928],{"className":12208,"style":4138},[61,62],[17,12210,12212],{"className":12211},[433],[17,12213,12215,12235],{"className":12214},[437,1313],[17,12216,12218,12232],{"className":12217},[441],[17,12219,12221],{"className":12220,"style":1404},[445],[17,12222,12223,12226],{"style":4153},[17,12224],{"className":12225,"style":453},[452],[17,12227,12229],{"className":12228},[457,458,459,460],[17,12230,410],{"className":12231},[61,460],[17,12233,1346],{"className":12234},[1345],[17,12236,12238],{"className":12237},[441],[17,12239,12241],{"className":12240,"style":1429},[445],[17,12242],{},[17,12244],{"className":12245,"style":372},[228],[17,12247,484],{"className":12248},[376],[17,12250],{"className":12251,"style":372},[228],[17,12253,12255,12258,12261,12264,12267],{"className":12254},[52],[17,12256],{"className":12257,"style":12038},[56],[17,12259,3905],{"className":12260},[1301],[17,12262],{"className":12263,"style":372},[228],[17,12265,484],{"className":12266},[376],[17,12268],{"className":12269,"style":372},[228],[17,12271,12273,12276],{"className":12272},[52],[17,12274],{"className":12275,"style":3965},[56],[17,12277,12279,12282],{"className":12278},[61],[17,12280,3928],{"className":12281,"style":4138},[61,62],[17,12283,12285],{"className":12284},[433],[17,12286,12288,12308],{"className":12287},[437,1313],[17,12289,12291,12305],{"className":12290},[441],[17,12292,12294],{"className":12293,"style":4097},[445],[17,12295,12296,12299],{"style":4153},[17,12297],{"className":12298,"style":453},[452],[17,12300,12302],{"className":12301},[457,458,459,460],[17,12303,39],{"className":12304,"style":63},[61,62,460],[17,12306,1346],{"className":12307},[1345],[17,12309,12311],{"className":12310},[441],[17,12312,12314],{"className":12313,"style":1429},[445],[17,12315],{},[10,12317,12318,12319,12347,12348,12376],{},"We rearrange the rows and columns of determinant ",[17,12320,12322,12335],{"className":12321,"translate":21},[20],[17,12323,12325],{"className":12324},[25],[27,12326,12327],{"xmlns":29},[31,12328,12329,12333],{},[34,12330,12331],{},[37,12332,111],{},[41,12334,111],{"encoding":43},[17,12336,12338],{"className":12337,"ariaHidden":48},[47],[17,12339,12341,12344],{"className":12340},[52],[17,12342],{"className":12343,"style":123},[56],[17,12345,111],{"className":12346,"style":127},[61,62]," such that ",[17,12349,12351,12364],{"className":12350,"translate":21},[20],[17,12352,12354],{"className":12353},[25],[27,12355,12356],{"xmlns":29},[31,12357,12358,12362],{},[34,12359,12360],{},[37,12361,172],{},[41,12363,172],{"encoding":43},[17,12365,12367],{"className":12366,"ariaHidden":48},[47],[17,12368,12370,12373],{"className":12369},[52],[17,12371],{"className":12372,"style":123},[56],[17,12374,172],{"className":12375,"style":187},[61,62]," is moved to the top-left corner.",[10,12378,12379,12380,12450,12451,12666,12667,12760,12761,12830,12831,12901,12902,12994],{},"First, swap the ",[17,12381,12383,12401],{"className":12382,"translate":21},[20],[17,12384,12386],{"className":12385},[25],[27,12387,12388],{"xmlns":29},[31,12389,12390,12398],{},[34,12391,12392],{},[848,12393,12394,12396],{},[37,12395,3890],{},[408,12397,3544],{},[41,12399,12400],{"encoding":43},"i_1",[17,12402,12404],{"className":12403,"ariaHidden":48},[47],[17,12405,12407,12410],{"className":12406},[52],[17,12408],{"className":12409,"style":11927},[56],[17,12411,12413,12416],{"className":12412},[61],[17,12414,3890],{"className":12415},[61,62],[17,12417,12419],{"className":12418},[433],[17,12420,12422,12442],{"className":12421},[437,1313],[17,12423,12425,12439],{"className":12424},[441],[17,12426,12428],{"className":12427,"style":1404},[445],[17,12429,12430,12433],{"style":1407},[17,12431],{"className":12432,"style":453},[452],[17,12434,12436],{"className":12435},[457,458,459,460],[17,12437,3544],{"className":12438},[61,460],[17,12440,1346],{"className":12441},[1345],[17,12443,12445],{"className":12444},[441],[17,12446,12448],{"className":12447,"style":1429},[445],[17,12449],{},"-th row with rows ",[17,12452,12454,12502],{"className":12453,"translate":21},[20],[17,12455,12457],{"className":12456},[25],[27,12458,12459],{"xmlns":29},[31,12460,12461,12499],{},[34,12462,12463,12469,12471,12473,12475,12481,12483,12485,12487,12489,12491,12493,12495,12497],{},[848,12464,12465,12467],{},[37,12466,3890],{},[408,12468,3544],{},[204,12470,627],{},[408,12472,3544],{},[204,12474,1084],{"separator":48},[848,12476,12477,12479],{},[37,12478,3890],{},[408,12480,3544],{},[204,12482,627],{},[408,12484,410],{},[204,12486,1084],{"separator":48},[204,12488,3905],{},[3907,12490,3909],{},[204,12492,1084],{"separator":48},[408,12494,410],{},[204,12496,1084],{"separator":48},[408,12498,3544],{},[41,12500,12501],{"encoding":43},"i_1-1, i_1-2, \\cdots, 2, 1",[17,12503,12505,12560,12624],{"className":12504,"ariaHidden":48},[47],[17,12506,12508,12511,12551,12554,12557],{"className":12507},[52],[17,12509],{"className":12510,"style":11927},[56],[17,12512,12514,12517],{"className":12513},[61],[17,12515,3890],{"className":12516},[61,62],[17,12518,12520],{"className":12519},[433],[17,12521,12523,12543],{"className":12522},[437,1313],[17,12524,12526,12540],{"className":12525},[441],[17,12527,12529],{"className":12528,"style":1404},[445],[17,12530,12531,12534],{"style":1407},[17,12532],{"className":12533,"style":453},[452],[17,12535,12537],{"className":12536},[457,458,459,460],[17,12538,3544],{"className":12539},[61,460],[17,12541,1346],{"className":12542},[1345],[17,12544,12546],{"className":12545},[441],[17,12547,12549],{"className":12548,"style":1429},[445],[17,12550],{},[17,12552],{"className":12553,"style":229},[228],[17,12555,627],{"className":12556},[233],[17,12558],{"className":12559,"style":229},[228],[17,12561,12563,12566,12569,12572,12575,12615,12618,12621],{"className":12562},[52],[17,12564],{"className":12565,"style":3965},[56],[17,12567,3544],{"className":12568},[61],[17,12570,1084],{"className":12571},[2410],[17,12573],{"className":12574,"style":2406},[228],[17,12576,12578,12581],{"className":12577},[61],[17,12579,3890],{"className":12580},[61,62],[17,12582,12584],{"className":12583},[433],[17,12585,12587,12607],{"className":12586},[437,1313],[17,12588,12590,12604],{"className":12589},[441],[17,12591,12593],{"className":12592,"style":1404},[445],[17,12594,12595,12598],{"style":1407},[17,12596],{"className":12597,"style":453},[452],[17,12599,12601],{"className":12600},[457,458,459,460],[17,12602,3544],{"className":12603},[61,460],[17,12605,1346],{"className":12606},[1345],[17,12608,12610],{"className":12609},[441],[17,12611,12613],{"className":12612,"style":1429},[445],[17,12614],{},[17,12616],{"className":12617,"style":229},[228],[17,12619,627],{"className":12620},[233],[17,12622],{"className":12623,"style":229},[228],[17,12625,12627,12630,12633,12636,12639,12642,12645,12648,12651,12654,12657,12660,12663],{"className":12626},[52],[17,12628],{"className":12629,"style":9613},[56],[17,12631,410],{"className":12632},[61],[17,12634,1084],{"className":12635},[2410],[17,12637],{"className":12638,"style":2406},[228],[17,12640,3905],{"className":12641},[1301],[17,12643],{"className":12644,"style":2406},[228],[17,12646],{"className":12647,"style":2406},[228],[17,12649,1084],{"className":12650},[2410],[17,12652],{"className":12653,"style":2406},[228],[17,12655,410],{"className":12656},[61],[17,12658,1084],{"className":12659},[2410],[17,12661],{"className":12662,"style":2406},[228],[17,12664,3544],{"className":12665},[61],", which takes ",[17,12668,12670,12692],{"className":12669,"translate":21},[20],[17,12671,12673],{"className":12672},[25],[27,12674,12675],{"xmlns":29},[31,12676,12677,12689],{},[34,12678,12679,12685,12687],{},[848,12680,12681,12683],{},[37,12682,3890],{},[408,12684,3544],{},[204,12686,627],{},[408,12688,3544],{},[41,12690,12691],{"encoding":43},"i_1-1",[17,12693,12695,12750],{"className":12694,"ariaHidden":48},[47],[17,12696,12698,12701,12741,12744,12747],{"className":12697},[52],[17,12699],{"className":12700,"style":11927},[56],[17,12702,12704,12707],{"className":12703},[61],[17,12705,3890],{"className":12706},[61,62],[17,12708,12710],{"className":12709},[433],[17,12711,12713,12733],{"className":12712},[437,1313],[17,12714,12716,12730],{"className":12715},[441],[17,12717,12719],{"className":12718,"style":1404},[445],[17,12720,12721,12724],{"style":1407},[17,12722],{"className":12723,"style":453},[452],[17,12725,12727],{"className":12726},[457,458,459,460],[17,12728,3544],{"className":12729},[61,460],[17,12731,1346],{"className":12732},[1345],[17,12734,12736],{"className":12735},[441],[17,12737,12739],{"className":12738,"style":1429},[445],[17,12740],{},[17,12742],{"className":12743,"style":229},[228],[17,12745,627],{"className":12746},[233],[17,12748],{"className":12749,"style":229},[228],[17,12751,12753,12757],{"className":12752},[52],[17,12754],{"className":12755,"style":12756},[56],"height:0.6444em;",[17,12758,3544],{"className":12759},[61]," swaps to move the ",[17,12762,12764,12781],{"className":12763,"translate":21},[20],[17,12765,12767],{"className":12766},[25],[27,12768,12769],{"xmlns":29},[31,12770,12771,12779],{},[34,12772,12773],{},[848,12774,12775,12777],{},[37,12776,3890],{},[408,12778,3544],{},[41,12780,12400],{"encoding":43},[17,12782,12784],{"className":12783,"ariaHidden":48},[47],[17,12785,12787,12790],{"className":12786},[52],[17,12788],{"className":12789,"style":11927},[56],[17,12791,12793,12796],{"className":12792},[61],[17,12794,3890],{"className":12795},[61,62],[17,12797,12799],{"className":12798},[433],[17,12800,12802,12822],{"className":12801},[437,1313],[17,12803,12805,12819],{"className":12804},[441],[17,12806,12808],{"className":12807,"style":1404},[445],[17,12809,12810,12813],{"style":1407},[17,12811],{"className":12812,"style":453},[452],[17,12814,12816],{"className":12815},[457,458,459,460],[17,12817,3544],{"className":12818},[61,460],[17,12820,1346],{"className":12821},[1345],[17,12823,12825],{"className":12824},[441],[17,12826,12828],{"className":12827,"style":1429},[445],[17,12829],{},"-th row to the first row. Then, swap the ",[17,12832,12834,12852],{"className":12833,"translate":21},[20],[17,12835,12837],{"className":12836},[25],[27,12838,12839],{"xmlns":29},[31,12840,12841,12849],{},[34,12842,12843],{},[848,12844,12845,12847],{},[37,12846,3890],{},[408,12848,410],{},[41,12850,12851],{"encoding":43},"i_2",[17,12853,12855],{"className":12854,"ariaHidden":48},[47],[17,12856,12858,12861],{"className":12857},[52],[17,12859],{"className":12860,"style":11927},[56],[17,12862,12864,12867],{"className":12863},[61],[17,12865,3890],{"className":12866},[61,62],[17,12868,12870],{"className":12869},[433],[17,12871,12873,12893],{"className":12872},[437,1313],[17,12874,12876,12890],{"className":12875},[441],[17,12877,12879],{"className":12878,"style":1404},[445],[17,12880,12881,12884],{"style":1407},[17,12882],{"className":12883,"style":453},[452],[17,12885,12887],{"className":12886},[457,458,459,460],[17,12888,410],{"className":12889},[61,460],[17,12891,1346],{"className":12892},[1345],[17,12894,12896],{"className":12895},[441],[17,12897,12899],{"className":12898,"style":1429},[445],[17,12900],{},"-th row ",[17,12903,12905,12927],{"className":12904,"translate":21},[20],[17,12906,12908],{"className":12907},[25],[27,12909,12910],{"xmlns":29},[31,12911,12912,12924],{},[34,12913,12914,12920,12922],{},[848,12915,12916,12918],{},[37,12917,3890],{},[408,12919,410],{},[204,12921,627],{},[408,12923,410],{},[41,12925,12926],{"encoding":43},"i_2-2",[17,12928,12930,12985],{"className":12929,"ariaHidden":48},[47],[17,12931,12933,12936,12976,12979,12982],{"className":12932},[52],[17,12934],{"className":12935,"style":11927},[56],[17,12937,12939,12942],{"className":12938},[61],[17,12940,3890],{"className":12941},[61,62],[17,12943,12945],{"className":12944},[433],[17,12946,12948,12968],{"className":12947},[437,1313],[17,12949,12951,12965],{"className":12950},[441],[17,12952,12954],{"className":12953,"style":1404},[445],[17,12955,12956,12959],{"style":1407},[17,12957],{"className":12958,"style":453},[452],[17,12960,12962],{"className":12961},[457,458,459,460],[17,12963,410],{"className":12964},[61,460],[17,12966,1346],{"className":12967},[1345],[17,12969,12971],{"className":12970},[441],[17,12972,12974],{"className":12973,"style":1429},[445],[17,12975],{},[17,12977],{"className":12978,"style":229},[228],[17,12980,627],{"className":12981},[233],[17,12983],{"className":12984,"style":229},[228],[17,12986,12988,12991],{"className":12987},[52],[17,12989],{"className":12990,"style":12756},[56],[17,12992,410],{"className":12993},[61]," times to move it to the second row, and so on. In total, this requires",[17,12996,12998],{"className":12997,"translate":21},[807],[17,12999,13001,13115],{"className":13000,"translate":21},[20],[17,13002,13004],{"className":13003},[25],[27,13005,13006],{"xmlns":29,"display":816},[31,13007,13008,13112],{},[34,13009,13010,13012,13018,13020,13022,13024,13026,13028,13034,13036,13038,13040,13042,13044,13046,13048,13054,13056,13058,13060,13062,13064,13070,13072,13078,13080,13082,13084,13090,13092,13094,13096,13098,13100,13102,13104,13106,13108,13110],{},[204,13011,622],{"stretchy":621},[848,13013,13014,13016],{},[37,13015,3890],{},[408,13017,3544],{},[204,13019,627],{},[408,13021,3544],{},[204,13023,632],{"stretchy":621},[204,13025,3557],{},[204,13027,622],{"stretchy":621},[848,13029,13030,13032],{},[37,13031,3890],{},[408,13033,410],{},[204,13035,627],{},[408,13037,410],{},[204,13039,632],{"stretchy":621},[204,13041,3557],{},[204,13043,3905],{},[204,13045,3557],{},[204,13047,622],{"stretchy":621},[848,13049,13050,13052],{},[37,13051,3890],{},[37,13053,39],{},[204,13055,627],{},[37,13057,39],{},[204,13059,632],{"stretchy":621},[204,13061,825],{},[204,13063,622],{"stretchy":621},[848,13065,13066,13068],{},[37,13067,3890],{},[408,13069,3544],{},[204,13071,3557],{},[848,13073,13074,13076],{},[37,13075,3890],{},[408,13077,410],{},[204,13079,3557],{},[204,13081,3905],{},[204,13083,3557],{},[848,13085,13086,13088],{},[37,13087,3890],{},[37,13089,39],{},[204,13091,632],{"stretchy":621},[204,13093,627],{},[204,13095,622],{"stretchy":621},[408,13097,3544],{},[204,13099,3557],{},[408,13101,410],{},[204,13103,3557],{},[204,13105,3905],{},[204,13107,3557],{},[37,13109,39],{},[204,13111,632],{"stretchy":621},[41,13113,13114],{"encoding":43},"(i_1-1)+(i_2-2)+\\cdots+(i_k-k)=(i_1+i_2+\\cdots+i_k)-(1+2+\\cdots+k)",[17,13116,13118,13176,13197,13255,13276,13295,13353,13374,13432,13487,13505,13563,13584,13603,13621],{"className":13117,"ariaHidden":48},[47],[17,13119,13121,13124,13127,13167,13170,13173],{"className":13120},[52],[17,13122],{"className":13123,"style":645},[56],[17,13125,622],{"className":13126},[649],[17,13128,13130,13133],{"className":13129},[61],[17,13131,3890],{"className":13132},[61,62],[17,13134,13136],{"className":13135},[433],[17,13137,13139,13159],{"className":13138},[437,1313],[17,13140,13142,13156],{"className":13141},[441],[17,13143,13145],{"className":13144,"style":1404},[445],[17,13146,13147,13150],{"style":1407},[17,13148],{"className":13149,"style":453},[452],[17,13151,13153],{"className":13152},[457,458,459,460],[17,13154,3544],{"className":13155},[61,460],[17,13157,1346],{"className":13158},[1345],[17,13160,13162],{"className":13161},[441],[17,13163,13165],{"className":13164,"style":1429},[445],[17,13166],{},[17,13168],{"className":13169,"style":229},[228],[17,13171,627],{"className":13172},[233],[17,13174],{"className":13175,"style":229},[228],[17,13177,13179,13182,13185,13188,13191,13194],{"className":13178},[52],[17,13180],{"className":13181,"style":645},[56],[17,13183,3544],{"className":13184},[61],[17,13186,632],{"className":13187},[674],[17,13189],{"className":13190,"style":229},[228],[17,13192,3557],{"className":13193},[233],[17,13195],{"className":13196,"style":229},[228],[17,13198,13200,13203,13206,13246,13249,13252],{"className":13199},[52],[17,13201],{"className":13202,"style":645},[56],[17,13204,622],{"className":13205},[649],[17,13207,13209,13212],{"className":13208},[61],[17,13210,3890],{"className":13211},[61,62],[17,13213,13215],{"className":13214},[433],[17,13216,13218,13238],{"className":13217},[437,1313],[17,13219,13221,13235],{"className":13220},[441],[17,13222,13224],{"className":13223,"style":1404},[445],[17,13225,13226,13229],{"style":1407},[17,13227],{"className":13228,"style":453},[452],[17,13230,13232],{"className":13231},[457,458,459,460],[17,13233,410],{"className":13234},[61,460],[17,13236,1346],{"className":13237},[1345],[17,13239,13241],{"className":13240},[441],[17,13242,13244],{"className":13243,"style":1429},[445],[17,13245],{},[17,13247],{"className":13248,"style":229},[228],[17,13250,627],{"className":13251},[233],[17,13253],{"className":13254,"style":229},[228],[17,13256,13258,13261,13264,13267,13270,13273],{"className":13257},[52],[17,13259],{"className":13260,"style":645},[56],[17,13262,410],{"className":13263},[61],[17,13265,632],{"className":13266},[674],[17,13268],{"className":13269,"style":229},[228],[17,13271,3557],{"className":13272},[233],[17,13274],{"className":13275,"style":229},[228],[17,13277,13279,13283,13286,13289,13292],{"className":13278},[52],[17,13280],{"className":13281,"style":13282},[56],"height:0.6667em;vertical-align:-0.0833em;",[17,13284,3905],{"className":13285},[1301],[17,13287],{"className":13288,"style":229},[228],[17,13290,3557],{"className":13291},[233],[17,13293],{"className":13294,"style":229},[228],[17,13296,13298,13301,13304,13344,13347,13350],{"className":13297},[52],[17,13299],{"className":13300,"style":645},[56],[17,13302,622],{"className":13303},[649],[17,13305,13307,13310],{"className":13306},[61],[17,13308,3890],{"className":13309},[61,62],[17,13311,13313],{"className":13312},[433],[17,13314,13316,13336],{"className":13315},[437,1313],[17,13317,13319,13333],{"className":13318},[441],[17,13320,13322],{"className":13321,"style":4097},[445],[17,13323,13324,13327],{"style":1407},[17,13325],{"className":13326,"style":453},[452],[17,13328,13330],{"className":13329},[457,458,459,460],[17,13331,39],{"className":13332,"style":63},[61,62,460],[17,13334,1346],{"className":13335},[1345],[17,13337,13339],{"className":13338},[441],[17,13340,13342],{"className":13341,"style":1429},[445],[17,13343],{},[17,13345],{"className":13346,"style":229},[228],[17,13348,627],{"className":13349},[233],[17,13351],{"className":13352,"style":229},[228],[17,13354,13356,13359,13362,13365,13368,13371],{"className":13355},[52],[17,13357],{"className":13358,"style":645},[56],[17,13360,39],{"className":13361,"style":63},[61,62],[17,13363,632],{"className":13364},[674],[17,13366],{"className":13367,"style":372},[228],[17,13369,825],{"className":13370},[376],[17,13372],{"className":13373,"style":372},[228],[17,13375,13377,13380,13383,13423,13426,13429],{"className":13376},[52],[17,13378],{"className":13379,"style":645},[56],[17,13381,622],{"className":13382},[649],[17,13384,13386,13389],{"className":13385},[61],[17,13387,3890],{"className":13388},[61,62],[17,13390,13392],{"className":13391},[433],[17,13393,13395,13415],{"className":13394},[437,1313],[17,13396,13398,13412],{"className":13397},[441],[17,13399,13401],{"className":13400,"style":1404},[445],[17,13402,13403,13406],{"style":1407},[17,13404],{"className":13405,"style":453},[452],[17,13407,13409],{"className":13408},[457,458,459,460],[17,13410,3544],{"className":13411},[61,460],[17,13413,1346],{"className":13414},[1345],[17,13416,13418],{"className":13417},[441],[17,13419,13421],{"className":13420,"style":1429},[445],[17,13422],{},[17,13424],{"className":13425,"style":229},[228],[17,13427,3557],{"className":13428},[233],[17,13430],{"className":13431,"style":229},[228],[17,13433,13435,13438,13478,13481,13484],{"className":13434},[52],[17,13436],{"className":13437,"style":11927},[56],[17,13439,13441,13444],{"className":13440},[61],[17,13442,3890],{"className":13443},[61,62],[17,13445,13447],{"className":13446},[433],[17,13448,13450,13470],{"className":13449},[437,1313],[17,13451,13453,13467],{"className":13452},[441],[17,13454,13456],{"className":13455,"style":1404},[445],[17,13457,13458,13461],{"style":1407},[17,13459],{"className":13460,"style":453},[452],[17,13462,13464],{"className":13463},[457,458,459,460],[17,13465,410],{"className":13466},[61,460],[17,13468,1346],{"className":13469},[1345],[17,13471,13473],{"className":13472},[441],[17,13474,13476],{"className":13475,"style":1429},[445],[17,13477],{},[17,13479],{"className":13480,"style":229},[228],[17,13482,3557],{"className":13483},[233],[17,13485],{"className":13486,"style":229},[228],[17,13488,13490,13493,13496,13499,13502],{"className":13489},[52],[17,13491],{"className":13492,"style":13282},[56],[17,13494,3905],{"className":13495},[1301],[17,13497],{"className":13498,"style":229},[228],[17,13500,3557],{"className":13501},[233],[17,13503],{"className":13504,"style":229},[228],[17,13506,13508,13511,13551,13554,13557,13560],{"className":13507},[52],[17,13509],{"className":13510,"style":645},[56],[17,13512,13514,13517],{"className":13513},[61],[17,13515,3890],{"className":13516},[61,62],[17,13518,13520],{"className":13519},[433],[17,13521,13523,13543],{"className":13522},[437,1313],[17,13524,13526,13540],{"className":13525},[441],[17,13527,13529],{"className":13528,"style":4097},[445],[17,13530,13531,13534],{"style":1407},[17,13532],{"className":13533,"style":453},[452],[17,13535,13537],{"className":13536},[457,458,459,460],[17,13538,39],{"className":13539,"style":63},[61,62,460],[17,13541,1346],{"className":13542},[1345],[17,13544,13546],{"className":13545},[441],[17,13547,13549],{"className":13548,"style":1429},[445],[17,13550],{},[17,13552,632],{"className":13553},[674],[17,13555],{"className":13556,"style":229},[228],[17,13558,627],{"className":13559},[233],[17,13561],{"className":13562,"style":229},[228],[17,13564,13566,13569,13572,13575,13578,13581],{"className":13565},[52],[17,13567],{"className":13568,"style":645},[56],[17,13570,622],{"className":13571},[649],[17,13573,3544],{"className":13574},[61],[17,13576],{"className":13577,"style":229},[228],[17,13579,3557],{"className":13580},[233],[17,13582],{"className":13583,"style":229},[228],[17,13585,13587,13591,13594,13597,13600],{"className":13586},[52],[17,13588],{"className":13589,"style":13590},[56],"height:0.7278em;vertical-align:-0.0833em;",[17,13592,410],{"className":13593},[61],[17,13595],{"className":13596,"style":229},[228],[17,13598,3557],{"className":13599},[233],[17,13601],{"className":13602,"style":229},[228],[17,13604,13606,13609,13612,13615,13618],{"className":13605},[52],[17,13607],{"className":13608,"style":13282},[56],[17,13610,3905],{"className":13611},[1301],[17,13613],{"className":13614,"style":229},[228],[17,13616,3557],{"className":13617},[233],[17,13619],{"className":13620,"style":229},[228],[17,13622,13624,13627,13630],{"className":13623},[52],[17,13625],{"className":13626,"style":645},[56],[17,13628,39],{"className":13629,"style":63},[61,62],[17,13631,632],{"className":13632},[674],[10,13634,13635,13636,13834,13835,798],{},"row swaps to move rows ",[17,13637,13639,13678],{"className":13638,"translate":21},[20],[17,13640,13642],{"className":13641},[25],[27,13643,13644],{"xmlns":29},[31,13645,13646,13676],{},[34,13647,13648,13654,13656,13662,13664,13666,13668,13670],{},[848,13649,13650,13652],{},[37,13651,3890],{},[408,13653,3544],{},[204,13655,1084],{"separator":48},[848,13657,13658,13660],{},[37,13659,3890],{},[408,13661,410],{},[204,13663,1084],{"separator":48},[204,13665,3905],{},[3907,13667,3909],{},[204,13669,1084],{"separator":48},[848,13671,13672,13674],{},[37,13673,3890],{},[37,13675,39],{},[41,13677,11492],{"encoding":43},[17,13679,13681],{"className":13680,"ariaHidden":48},[47],[17,13682,13684,13687,13727,13730,13733,13773,13776,13779,13782,13785,13788,13791,13794],{"className":13683},[52],[17,13685],{"className":13686,"style":3965},[56],[17,13688,13690,13693],{"className":13689},[61],[17,13691,3890],{"className":13692},[61,62],[17,13694,13696],{"className":13695},[433],[17,13697,13699,13719],{"className":13698},[437,1313],[17,13700,13702,13716],{"className":13701},[441],[17,13703,13705],{"className":13704,"style":1404},[445],[17,13706,13707,13710],{"style":1407},[17,13708],{"className":13709,"style":453},[452],[17,13711,13713],{"className":13712},[457,458,459,460],[17,13714,3544],{"className":13715},[61,460],[17,13717,1346],{"className":13718},[1345],[17,13720,13722],{"className":13721},[441],[17,13723,13725],{"className":13724,"style":1429},[445],[17,13726],{},[17,13728,1084],{"className":13729},[2410],[17,13731],{"className":13732,"style":2406},[228],[17,13734,13736,13739],{"className":13735},[61],[17,13737,3890],{"className":13738},[61,62],[17,13740,13742],{"className":13741},[433],[17,13743,13745,13765],{"className":13744},[437,1313],[17,13746,13748,13762],{"className":13747},[441],[17,13749,13751],{"className":13750,"style":1404},[445],[17,13752,13753,13756],{"style":1407},[17,13754],{"className":13755,"style":453},[452],[17,13757,13759],{"className":13758},[457,458,459,460],[17,13760,410],{"className":13761},[61,460],[17,13763,1346],{"className":13764},[1345],[17,13766,13768],{"className":13767},[441],[17,13769,13771],{"className":13770,"style":1429},[445],[17,13772],{},[17,13774,1084],{"className":13775},[2410],[17,13777],{"className":13778,"style":2406},[228],[17,13780,3905],{"className":13781},[1301],[17,13783],{"className":13784,"style":2406},[228],[17,13786],{"className":13787,"style":2406},[228],[17,13789,1084],{"className":13790},[2410],[17,13792],{"className":13793,"style":2406},[228],[17,13795,13797,13800],{"className":13796},[61],[17,13798,3890],{"className":13799},[61,62],[17,13801,13803],{"className":13802},[433],[17,13804,13806,13826],{"className":13805},[437,1313],[17,13807,13809,13823],{"className":13808},[441],[17,13810,13812],{"className":13811,"style":4097},[445],[17,13813,13814,13817],{"style":1407},[17,13815],{"className":13816,"style":453},[452],[17,13818,13820],{"className":13819},[457,458,459,460],[17,13821,39],{"className":13822,"style":63},[61,62,460],[17,13824,1346],{"className":13825},[1345],[17,13827,13829],{"className":13828},[441],[17,13830,13832],{"className":13831,"style":1429},[445],[17,13833],{}," to rows ",[17,13836,13838,13866],{"className":13837,"translate":21},[20],[17,13839,13841],{"className":13840},[25],[27,13842,13843],{"xmlns":29},[31,13844,13845,13863],{},[34,13846,13847,13849,13851,13853,13855,13857,13859,13861],{},[408,13848,3544],{},[204,13850,1084],{"separator":48},[408,13852,410],{},[204,13854,1084],{"separator":48},[204,13856,3905],{},[3907,13858,3909],{},[204,13860,1084],{"separator":48},[37,13862,39],{},[41,13864,13865],{"encoding":43},"1, 2, \\cdots, k",[17,13867,13869],{"className":13868,"ariaHidden":48},[47],[17,13870,13872,13875,13878,13881,13884,13887,13890,13893,13896,13899,13902,13905,13908],{"className":13871},[52],[17,13873],{"className":13874,"style":8579},[56],[17,13876,3544],{"className":13877},[61],[17,13879,1084],{"className":13880},[2410],[17,13882],{"className":13883,"style":2406},[228],[17,13885,410],{"className":13886},[61],[17,13888,1084],{"className":13889},[2410],[17,13891],{"className":13892,"style":2406},[228],[17,13894,3905],{"className":13895},[1301],[17,13897],{"className":13898,"style":2406},[228],[17,13900],{"className":13901,"style":2406},[228],[17,13903,1084],{"className":13904},[2410],[17,13906],{"className":13907,"style":2406},[228],[17,13909,39],{"className":13910,"style":63},[61,62],[10,13912,13913,13914,13942,13943,14018],{},"Using similar column swaps, we move the columns of ",[17,13915,13917,13930],{"className":13916,"translate":21},[20],[17,13918,13920],{"className":13919},[25],[27,13921,13922],{"xmlns":29},[31,13923,13924,13928],{},[34,13925,13926],{},[37,13927,172],{},[41,13929,172],{"encoding":43},[17,13931,13933],{"className":13932,"ariaHidden":48},[47],[17,13934,13936,13939],{"className":13935},[52],[17,13937],{"className":13938,"style":123},[56],[17,13940,172],{"className":13941,"style":187},[61,62]," to columns ",[17,13944,13946,13973],{"className":13945,"translate":21},[20],[17,13947,13949],{"className":13948},[25],[27,13950,13951],{"xmlns":29},[31,13952,13953,13971],{},[34,13954,13955,13957,13959,13961,13963,13965,13967,13969],{},[408,13956,3544],{},[204,13958,1084],{"separator":48},[408,13960,410],{},[204,13962,1084],{"separator":48},[204,13964,3905],{},[3907,13966,3909],{},[204,13968,1084],{"separator":48},[37,13970,39],{},[41,13972,13865],{"encoding":43},[17,13974,13976],{"className":13975,"ariaHidden":48},[47],[17,13977,13979,13982,13985,13988,13991,13994,13997,14000,14003,14006,14009,14012,14015],{"className":13978},[52],[17,13980],{"className":13981,"style":8579},[56],[17,13983,3544],{"className":13984},[61],[17,13986,1084],{"className":13987},[2410],[17,13989],{"className":13990,"style":2406},[228],[17,13992,410],{"className":13993},[61],[17,13995,1084],{"className":13996},[2410],[17,13998],{"className":13999,"style":2406},[228],[17,14001,3905],{"className":14002},[1301],[17,14004],{"className":14005,"style":2406},[228],[17,14007],{"className":14008,"style":2406},[228],[17,14010,1084],{"className":14011},[2410],[17,14013],{"className":14014,"style":2406},[228],[17,14016,39],{"className":14017,"style":63},[61,62],", requiring",[17,14020,14022],{"className":14021,"translate":21},[807],[17,14023,14025,14139],{"className":14024,"translate":21},[20],[17,14026,14028],{"className":14027},[25],[27,14029,14030],{"xmlns":29,"display":816},[31,14031,14032,14136],{},[34,14033,14034,14036,14042,14044,14046,14048,14050,14052,14058,14060,14062,14064,14066,14068,14070,14072,14078,14080,14082,14084,14086,14088,14094,14096,14102,14104,14106,14108,14114,14116,14118,14120,14122,14124,14126,14128,14130,14132,14134],{},[204,14035,622],{"stretchy":621},[848,14037,14038,14040],{},[37,14039,3928],{},[408,14041,3544],{},[204,14043,627],{},[408,14045,3544],{},[204,14047,632],{"stretchy":621},[204,14049,3557],{},[204,14051,622],{"stretchy":621},[848,14053,14054,14056],{},[37,14055,3928],{},[408,14057,410],{},[204,14059,627],{},[408,14061,410],{},[204,14063,632],{"stretchy":621},[204,14065,3557],{},[204,14067,3905],{},[204,14069,3557],{},[204,14071,622],{"stretchy":621},[848,14073,14074,14076],{},[37,14075,3928],{},[37,14077,39],{},[204,14079,627],{},[37,14081,39],{},[204,14083,632],{"stretchy":621},[204,14085,825],{},[204,14087,622],{"stretchy":621},[848,14089,14090,14092],{},[37,14091,3928],{},[408,14093,3544],{},[204,14095,3557],{},[848,14097,14098,14100],{},[37,14099,3928],{},[408,14101,410],{},[204,14103,3557],{},[204,14105,3905],{},[204,14107,3557],{},[848,14109,14110,14112],{},[37,14111,3928],{},[37,14113,39],{},[204,14115,632],{"stretchy":621},[204,14117,627],{},[204,14119,622],{"stretchy":621},[408,14121,3544],{},[204,14123,3557],{},[408,14125,410],{},[204,14127,3557],{},[204,14129,3905],{},[204,14131,3557],{},[37,14133,39],{},[204,14135,632],{"stretchy":621},[41,14137,14138],{"encoding":43},"(j_1-1)+(j_2-2)+\\cdots+(j_k-k)=(j_1+j_2+\\cdots+j_k)-(1+2+\\cdots+k)",[17,14140,14142,14200,14221,14279,14300,14318,14376,14397,14455,14510,14528,14586,14607,14625,14643],{"className":14141,"ariaHidden":48},[47],[17,14143,14145,14148,14151,14191,14194,14197],{"className":14144},[52],[17,14146],{"className":14147,"style":645},[56],[17,14149,622],{"className":14150},[649],[17,14152,14154,14157],{"className":14153},[61],[17,14155,3928],{"className":14156,"style":4138},[61,62],[17,14158,14160],{"className":14159},[433],[17,14161,14163,14183],{"className":14162},[437,1313],[17,14164,14166,14180],{"className":14165},[441],[17,14167,14169],{"className":14168,"style":1404},[445],[17,14170,14171,14174],{"style":4153},[17,14172],{"className":14173,"style":453},[452],[17,14175,14177],{"className":14176},[457,458,459,460],[17,14178,3544],{"className":14179},[61,460],[17,14181,1346],{"className":14182},[1345],[17,14184,14186],{"className":14185},[441],[17,14187,14189],{"className":14188,"style":1429},[445],[17,14190],{},[17,14192],{"className":14193,"style":229},[228],[17,14195,627],{"className":14196},[233],[17,14198],{"className":14199,"style":229},[228],[17,14201,14203,14206,14209,14212,14215,14218],{"className":14202},[52],[17,14204],{"className":14205,"style":645},[56],[17,14207,3544],{"className":14208},[61],[17,14210,632],{"className":14211},[674],[17,14213],{"className":14214,"style":229},[228],[17,14216,3557],{"className":14217},[233],[17,14219],{"className":14220,"style":229},[228],[17,14222,14224,14227,14230,14270,14273,14276],{"className":14223},[52],[17,14225],{"className":14226,"style":645},[56],[17,14228,622],{"className":14229},[649],[17,14231,14233,14236],{"className":14232},[61],[17,14234,3928],{"className":14235,"style":4138},[61,62],[17,14237,14239],{"className":14238},[433],[17,14240,14242,14262],{"className":14241},[437,1313],[17,14243,14245,14259],{"className":14244},[441],[17,14246,14248],{"className":14247,"style":1404},[445],[17,14249,14250,14253],{"style":4153},[17,14251],{"className":14252,"style":453},[452],[17,14254,14256],{"className":14255},[457,458,459,460],[17,14257,410],{"className":14258},[61,460],[17,14260,1346],{"className":14261},[1345],[17,14263,14265],{"className":14264},[441],[17,14266,14268],{"className":14267,"style":1429},[445],[17,14269],{},[17,14271],{"className":14272,"style":229},[228],[17,14274,627],{"className":14275},[233],[17,14277],{"className":14278,"style":229},[228],[17,14280,14282,14285,14288,14291,14294,14297],{"className":14281},[52],[17,14283],{"className":14284,"style":645},[56],[17,14286,410],{"className":14287},[61],[17,14289,632],{"className":14290},[674],[17,14292],{"className":14293,"style":229},[228],[17,14295,3557],{"className":14296},[233],[17,14298],{"className":14299,"style":229},[228],[17,14301,14303,14306,14309,14312,14315],{"className":14302},[52],[17,14304],{"className":14305,"style":13282},[56],[17,14307,3905],{"className":14308},[1301],[17,14310],{"className":14311,"style":229},[228],[17,14313,3557],{"className":14314},[233],[17,14316],{"className":14317,"style":229},[228],[17,14319,14321,14324,14327,14367,14370,14373],{"className":14320},[52],[17,14322],{"className":14323,"style":645},[56],[17,14325,622],{"className":14326},[649],[17,14328,14330,14333],{"className":14329},[61],[17,14331,3928],{"className":14332,"style":4138},[61,62],[17,14334,14336],{"className":14335},[433],[17,14337,14339,14359],{"className":14338},[437,1313],[17,14340,14342,14356],{"className":14341},[441],[17,14343,14345],{"className":14344,"style":4097},[445],[17,14346,14347,14350],{"style":4153},[17,14348],{"className":14349,"style":453},[452],[17,14351,14353],{"className":14352},[457,458,459,460],[17,14354,39],{"className":14355,"style":63},[61,62,460],[17,14357,1346],{"className":14358},[1345],[17,14360,14362],{"className":14361},[441],[17,14363,14365],{"className":14364,"style":1429},[445],[17,14366],{},[17,14368],{"className":14369,"style":229},[228],[17,14371,627],{"className":14372},[233],[17,14374],{"className":14375,"style":229},[228],[17,14377,14379,14382,14385,14388,14391,14394],{"className":14378},[52],[17,14380],{"className":14381,"style":645},[56],[17,14383,39],{"className":14384,"style":63},[61,62],[17,14386,632],{"className":14387},[674],[17,14389],{"className":14390,"style":372},[228],[17,14392,825],{"className":14393},[376],[17,14395],{"className":14396,"style":372},[228],[17,14398,14400,14403,14406,14446,14449,14452],{"className":14399},[52],[17,14401],{"className":14402,"style":645},[56],[17,14404,622],{"className":14405},[649],[17,14407,14409,14412],{"className":14408},[61],[17,14410,3928],{"className":14411,"style":4138},[61,62],[17,14413,14415],{"className":14414},[433],[17,14416,14418,14438],{"className":14417},[437,1313],[17,14419,14421,14435],{"className":14420},[441],[17,14422,14424],{"className":14423,"style":1404},[445],[17,14425,14426,14429],{"style":4153},[17,14427],{"className":14428,"style":453},[452],[17,14430,14432],{"className":14431},[457,458,459,460],[17,14433,3544],{"className":14434},[61,460],[17,14436,1346],{"className":14437},[1345],[17,14439,14441],{"className":14440},[441],[17,14442,14444],{"className":14443,"style":1429},[445],[17,14445],{},[17,14447],{"className":14448,"style":229},[228],[17,14450,3557],{"className":14451},[233],[17,14453],{"className":14454,"style":229},[228],[17,14456,14458,14461,14501,14504,14507],{"className":14457},[52],[17,14459],{"className":14460,"style":3965},[56],[17,14462,14464,14467],{"className":14463},[61],[17,14465,3928],{"className":14466,"style":4138},[61,62],[17,14468,14470],{"className":14469},[433],[17,14471,14473,14493],{"className":14472},[437,1313],[17,14474,14476,14490],{"className":14475},[441],[17,14477,14479],{"className":14478,"style":1404},[445],[17,14480,14481,14484],{"style":4153},[17,14482],{"className":14483,"style":453},[452],[17,14485,14487],{"className":14486},[457,458,459,460],[17,14488,410],{"className":14489},[61,460],[17,14491,1346],{"className":14492},[1345],[17,14494,14496],{"className":14495},[441],[17,14497,14499],{"className":14498,"style":1429},[445],[17,14500],{},[17,14502],{"className":14503,"style":229},[228],[17,14505,3557],{"className":14506},[233],[17,14508],{"className":14509,"style":229},[228],[17,14511,14513,14516,14519,14522,14525],{"className":14512},[52],[17,14514],{"className":14515,"style":13282},[56],[17,14517,3905],{"className":14518},[1301],[17,14520],{"className":14521,"style":229},[228],[17,14523,3557],{"className":14524},[233],[17,14526],{"className":14527,"style":229},[228],[17,14529,14531,14534,14574,14577,14580,14583],{"className":14530},[52],[17,14532],{"className":14533,"style":645},[56],[17,14535,14537,14540],{"className":14536},[61],[17,14538,3928],{"className":14539,"style":4138},[61,62],[17,14541,14543],{"className":14542},[433],[17,14544,14546,14566],{"className":14545},[437,1313],[17,14547,14549,14563],{"className":14548},[441],[17,14550,14552],{"className":14551,"style":4097},[445],[17,14553,14554,14557],{"style":4153},[17,14555],{"className":14556,"style":453},[452],[17,14558,14560],{"className":14559},[457,458,459,460],[17,14561,39],{"className":14562,"style":63},[61,62,460],[17,14564,1346],{"className":14565},[1345],[17,14567,14569],{"className":14568},[441],[17,14570,14572],{"className":14571,"style":1429},[445],[17,14573],{},[17,14575,632],{"className":14576},[674],[17,14578],{"className":14579,"style":229},[228],[17,14581,627],{"className":14582},[233],[17,14584],{"className":14585,"style":229},[228],[17,14587,14589,14592,14595,14598,14601,14604],{"className":14588},[52],[17,14590],{"className":14591,"style":645},[56],[17,14593,622],{"className":14594},[649],[17,14596,3544],{"className":14597},[61],[17,14599],{"className":14600,"style":229},[228],[17,14602,3557],{"className":14603},[233],[17,14605],{"className":14606,"style":229},[228],[17,14608,14610,14613,14616,14619,14622],{"className":14609},[52],[17,14611],{"className":14612,"style":13590},[56],[17,14614,410],{"className":14615},[61],[17,14617],{"className":14618,"style":229},[228],[17,14620,3557],{"className":14621},[233],[17,14623],{"className":14624,"style":229},[228],[17,14626,14628,14631,14634,14637,14640],{"className":14627},[52],[17,14629],{"className":14630,"style":13282},[56],[17,14632,3905],{"className":14633},[1301],[17,14635],{"className":14636,"style":229},[228],[17,14638,3557],{"className":14639},[233],[17,14641],{"className":14642,"style":229},[228],[17,14644,14646,14649,14652],{"className":14645},[52],[17,14647],{"className":14648,"style":645},[56],[17,14650,39],{"className":14651,"style":63},[61,62],[17,14653,632],{"className":14654},[674],[10,14656,14657],{},"column swaps.",[10,14659,14660,14661,14733],{},"Let ",[17,14662,14664,14682],{"className":14663,"translate":21},[20],[17,14665,14667],{"className":14666},[25],[27,14668,14669],{"xmlns":29},[31,14670,14671,14679],{},[34,14672,14673],{},[848,14674,14675,14677],{},[37,14676,111],{},[408,14678,3544],{},[41,14680,14681],{"encoding":43},"D_1",[17,14683,14685],{"className":14684,"ariaHidden":48},[47],[17,14686,14688,14692],{"className":14687},[52],[17,14689],{"className":14690,"style":14691},[56],"height:0.8333em;vertical-align:-0.15em;",[17,14693,14695,14698],{"className":14694},[61],[17,14696,111],{"className":14697,"style":127},[61,62],[17,14699,14701],{"className":14700},[433],[17,14702,14704,14725],{"className":14703},[437,1313],[17,14705,14707,14722],{"className":14706},[441],[17,14708,14710],{"className":14709,"style":1404},[445],[17,14711,14713,14716],{"style":14712},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[17,14714],{"className":14715,"style":453},[452],[17,14717,14719],{"className":14718},[457,458,459,460],[17,14720,3544],{"className":14721},[61,460],[17,14723,1346],{"className":14724},[1345],[17,14726,14728],{"className":14727},[441],[17,14729,14731],{"className":14730,"style":1429},[445],[17,14732],{}," denote the new determinant after these transformations. We have:",[17,14735,14737],{"className":14736,"translate":21},[807],[17,14738,14740,14948],{"className":14739,"translate":21},[20],[17,14741,14743],{"className":14742},[25],[27,14744,14745],{"xmlns":29,"display":816},[31,14746,14747,14945],{},[34,14748,14749,14755,14757,14759,14761,14763,14871,14873,14875,14877,14879,14881,14883,14943],{},[848,14750,14751,14753],{},[37,14752,111],{},[408,14754,3544],{},[204,14756,825],{},[204,14758,622],{"stretchy":621},[204,14760,627],{},[408,14762,3544],{},[403,14764,14765,14767],{},[204,14766,632],{"stretchy":621},[34,14768,14769,14771,14777,14779,14785,14787,14789,14791,14797,14799,14801,14803,14805,14807,14809,14811,14813,14815,14817,14819,14821,14823,14829,14831,14837,14839,14841,14843,14849,14851,14853,14855,14857,14859,14861,14863,14865,14867,14869],{},[204,14770,622],{"stretchy":621},[848,14772,14773,14775],{},[37,14774,3890],{},[408,14776,3544],{},[204,14778,3557],{},[848,14780,14781,14783],{},[37,14782,3890],{},[408,14784,410],{},[204,14786,3557],{},[204,14788,3905],{},[204,14790,3557],{},[848,14792,14793,14795],{},[37,14794,3890],{},[37,14796,39],{},[204,14798,632],{"stretchy":621},[204,14800,627],{},[204,14802,622],{"stretchy":621},[408,14804,3544],{},[204,14806,3557],{},[408,14808,410],{},[204,14810,3557],{},[204,14812,3905],{},[204,14814,3557],{},[37,14816,39],{},[204,14818,632],{"stretchy":621},[204,14820,3557],{},[204,14822,622],{"stretchy":621},[848,14824,14825,14827],{},[37,14826,3928],{},[408,14828,3544],{},[204,14830,3557],{},[848,14832,14833,14835],{},[37,14834,3928],{},[408,14836,410],{},[204,14838,3557],{},[204,14840,3905],{},[204,14842,3557],{},[848,14844,14845,14847],{},[37,14846,3928],{},[37,14848,39],{},[204,14850,632],{"stretchy":621},[204,14852,627],{},[204,14854,622],{"stretchy":621},[408,14856,3544],{},[204,14858,3557],{},[408,14860,410],{},[204,14862,3557],{},[204,14864,3905],{},[204,14866,3557],{},[37,14868,39],{},[204,14870,632],{"stretchy":621},[37,14872,111],{},[228,14874],{"linebreak":1087},[204,14876,825],{},[204,14878,622],{"stretchy":621},[204,14880,627],{},[408,14882,3544],{},[403,14884,14885,14887],{},[204,14886,632],{"stretchy":621},[34,14888,14889,14895,14897,14903,14905,14907,14909,14915,14917,14923,14925,14931,14933,14935,14937],{},[848,14890,14891,14893],{},[37,14892,3890],{},[408,14894,3544],{},[204,14896,3557],{},[848,14898,14899,14901],{},[37,14900,3890],{},[408,14902,410],{},[204,14904,3557],{},[204,14906,3905],{},[204,14908,3557],{},[848,14910,14911,14913],{},[37,14912,3890],{},[37,14914,39],{},[204,14916,3557],{},[848,14918,14919,14921],{},[37,14920,3928],{},[408,14922,3544],{},[204,14924,3557],{},[848,14926,14927,14929],{},[37,14928,3928],{},[408,14930,410],{},[204,14932,3557],{},[204,14934,3905],{},[204,14936,3557],{},[848,14938,14939,14941],{},[37,14940,3928],{},[37,14942,39],{},[37,14944,111],{},[41,14946,14947],{"encoding":43},"D_1 = (-1)^{(i_1+i_2+\\cdots+i_k)-(1+2+\\cdots+k)+(j_1+j_2+\\cdots+j_k)-(1+2+\\cdots+k)}D \\\\ = (-1)^{i_1+i_2+\\cdots+i_k+j_1+j_2+\\cdots+j_k}D",[17,14949,14951,15006,15392,15395,15408],{"className":14950,"ariaHidden":48},[47],[17,14952,14954,14957,14997,15000,15003],{"className":14953},[52],[17,14955],{"className":14956,"style":14691},[56],[17,14958,14960,14963],{"className":14959},[61],[17,14961,111],{"className":14962,"style":127},[61,62],[17,14964,14966],{"className":14965},[433],[17,14967,14969,14989],{"className":14968},[437,1313],[17,14970,14972,14986],{"className":14971},[441],[17,14973,14975],{"className":14974,"style":1404},[445],[17,14976,14977,14980],{"style":14712},[17,14978],{"className":14979,"style":453},[452],[17,14981,14983],{"className":14982},[457,458,459,460],[17,14984,3544],{"className":14985},[61,460],[17,14987,1346],{"className":14988},[1345],[17,14990,14992],{"className":14991},[441],[17,14993,14995],{"className":14994,"style":1429},[445],[17,14996],{},[17,14998],{"className":14999,"style":372},[228],[17,15001,825],{"className":15002},[376],[17,15004],{"className":15005,"style":372},[228],[17,15007,15009,15012,15015,15018,15021,15389],{"className":15008},[52],[17,15010],{"className":15011,"style":3613},[56],[17,15013,622],{"className":15014},[649],[17,15016,627],{"className":15017},[61],[17,15019,3544],{"className":15020},[61],[17,15022,15024,15027],{"className":15023},[674],[17,15025,632],{"className":15026},[674],[17,15028,15030],{"className":15029},[433],[17,15031,15033],{"className":15032},[437],[17,15034,15036],{"className":15035},[441],[17,15037,15039],{"className":15038,"style":3641},[445],[17,15040,15041,15044],{"style":3110},[17,15042],{"className":15043,"style":453},[452],[17,15045,15047],{"className":15046},[457,458,459,460],[17,15048,15050,15053,15093,15096,15136,15139,15142,15145,15185,15188,15191,15194,15197,15200,15203,15206,15209,15212,15215,15218,15221,15224,15264,15267,15307,15310,15313,15316,15356,15359,15362,15365,15368,15371,15374,15377,15380,15383,15386],{"className":15049},[61,460],[17,15051,622],{"className":15052},[649,460],[17,15054,15056,15059],{"className":15055},[61,460],[17,15057,3890],{"className":15058},[61,62,460],[17,15060,15062],{"className":15061},[433],[17,15063,15065,15085],{"className":15064},[437,1313],[17,15066,15068,15082],{"className":15067},[441],[17,15069,15071],{"className":15070,"style":4541},[445],[17,15072,15073,15076],{"style":4544},[17,15074],{"className":15075,"style":4548},[452],[17,15077,15079],{"className":15078},[457,4552,4553,460],[17,15080,3544],{"className":15081},[61,460],[17,15083,1346],{"className":15084},[1345],[17,15086,15088],{"className":15087},[441],[17,15089,15091],{"className":15090,"style":4569},[445],[17,15092],{},[17,15094,3557],{"className":15095},[233,460],[17,15097,15099,15102],{"className":15098},[61,460],[17,15100,3890],{"className":15101},[61,62,460],[17,15103,15105],{"className":15104},[433],[17,15106,15108,15128],{"className":15107},[437,1313],[17,15109,15111,15125],{"className":15110},[441],[17,15112,15114],{"className":15113,"style":4541},[445],[17,15115,15116,15119],{"style":4544},[17,15117],{"className":15118,"style":4548},[452],[17,15120,15122],{"className":15121},[457,4552,4553,460],[17,15123,410],{"className":15124},[61,460],[17,15126,1346],{"className":15127},[1345],[17,15129,15131],{"className":15130},[441],[17,15132,15134],{"className":15133,"style":4569},[445],[17,15135],{},[17,15137,3557],{"className":15138},[233,460],[17,15140,3905],{"className":15141},[1301,460],[17,15143,3557],{"className":15144},[233,460],[17,15146,15148,15151],{"className":15147},[61,460],[17,15149,3890],{"className":15150},[61,62,460],[17,15152,15154],{"className":15153},[433],[17,15155,15157,15177],{"className":15156},[437,1313],[17,15158,15160,15174],{"className":15159},[441],[17,15161,15163],{"className":15162,"style":4645},[445],[17,15164,15165,15168],{"style":4648},[17,15166],{"className":15167,"style":4548},[452],[17,15169,15171],{"className":15170},[457,4552,4553,460],[17,15172,39],{"className":15173,"style":63},[61,62,460],[17,15175,1346],{"className":15176},[1345],[17,15178,15180],{"className":15179},[441],[17,15181,15183],{"className":15182,"style":4670},[445],[17,15184],{},[17,15186,632],{"className":15187},[674,460],[17,15189,627],{"className":15190},[233,460],[17,15192,622],{"className":15193},[649,460],[17,15195,3544],{"className":15196},[61,460],[17,15198,3557],{"className":15199},[233,460],[17,15201,410],{"className":15202},[61,460],[17,15204,3557],{"className":15205},[233,460],[17,15207,3905],{"className":15208},[1301,460],[17,15210,3557],{"className":15211},[233,460],[17,15213,39],{"className":15214,"style":63},[61,62,460],[17,15216,632],{"className":15217},[674,460],[17,15219,3557],{"className":15220},[233,460],[17,15222,622],{"className":15223},[649,460],[17,15225,15227,15230],{"className":15226},[61,460],[17,15228,3928],{"className":15229,"style":4138},[61,62,460],[17,15231,15233],{"className":15232},[433],[17,15234,15236,15256],{"className":15235},[437,1313],[17,15237,15239,15253],{"className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15541,3905],{"className":15542},[1301,460],[17,15544,3557],{"className":15545},[233,460],[17,15547,15549,15552],{"className":15548},[61,460],[17,15550,3890],{"className":15551},[61,62,460],[17,15553,15555],{"className":15554},[433],[17,15556,15558,15578],{"className":15557},[437,1313],[17,15559,15561,15575],{"className":15560},[441],[17,15562,15564],{"className":15563,"style":4645},[445],[17,15565,15566,15569],{"style":4648},[17,15567],{"className":15568,"style":4548},[452],[17,15570,15572],{"className":15571},[457,4552,4553,460],[17,15573,39],{"className":15574,"style":63},[61,62,460],[17,15576,1346],{"className":15577},[1345],[17,15579,15581],{"className":15580},[441],[17,15582,15584],{"className":15583,"style":4670},[445],[17,15585],{},[17,15587,3557],{"className":15588},[233,460],[17,15590,15592,15595],{"className":15591},[61,460],[17,15593,3928],{"className":15594,"style":4138},[61,62,460],[17,15596,15598],{"className":15597},[433],[17,15599,15601,15621],{"className":15600},[437,1313],[17,15602,15604,15618],{"className":15603},[441],[17,15605,15607],{"className":15606,"style":4541},[445],[17,15608,15609,15612],{"style":4702},[17,15610],{"className":15611,"style":4548},[452],[17,15613,15615],{"className":15614},[457,4552,4553,460],[17,15616,3544],{"className":15617},[61,460],[17,15619,1346],{"className":15620},[1345],[17,15622,15624],{"className":15623},[441],[17,15625,15627],{"className":15626,"style":4569},[445],[17,15628],{},[17,15630,3557],{"className":15631},[233,460],[17,15633,15635,15638],{"className":15634},[61,460],[17,15636,3928],{"className":15637,"style":4138},[61,62,460],[17,15639,15641],{"className":15640},[433],[17,15642,15644,15664],{"className":15643},[437,1313],[17,15645,15647,15661],{"className":15646},[441],[17,15648,15650],{"className":15649,"style":4541},[445],[17,15651,15652,15655],{"style":4702},[17,15653],{"className":15654,"style":4548},[452],[17,15656,15658],{"className":15657},[457,4552,4553,460],[17,15659,410],{"className":15660},[61,460],[17,15662,1346],{"className":15663},[1345],[17,15665,15667],{"className":15666},[441],[17,15668,15670],{"className":15669,"style":4569},[445],[17,15671],{},[17,15673,3557],{"className":15674},[233,460],[17,15676,3905],{"className":15677},[1301,460],[17,15679,3557],{"className":15680},[233,460],[17,15682,15684,15687],{"className":15683},[61,460],[17,15685,3928],{"className":15686,"style":4138},[61,62,460],[17,15688,15690],{"className":15689},[433],[17,15691,15693,15713],{"className":15692},[437,1313],[17,15694,15696,15710],{"className":15695},[441],[17,15697,15699],{"className":15698,"style":4645},[445],[17,15700,15701,15704],{"style":4801},[17,15702],{"className":15703,"style":4548},[452],[17,15705,15707],{"className":15706},[457,4552,4553,460],[17,15708,39],{"className":15709,"style":63},[61,62,460],[17,15711,1346],{"className":15712},[1345],[17,15714,15716],{"className":15715},[441],[17,15717,15719],{"className":15718,"style":4670},[445],[17,15720],{},[17,15722,111],{"className":15723,"style":127},[61,62],[10,15725,15726,15727,15796,15797,15825,15826,798],{},"Thus, the expansion of ",[17,15728,15730,15747],{"className":15729,"translate":21},[20],[17,15731,15733],{"className":15732},[25],[27,15734,15735],{"xmlns":29},[31,15736,15737,15745],{},[34,15738,15739],{},[848,15740,15741,15743],{},[37,15742,111],{},[408,15744,3544],{},[41,15746,14681],{"encoding":43},[17,15748,15750],{"className":15749,"ariaHidden":48},[47],[17,15751,15753,15756],{"className":15752},[52],[17,15754],{"className":15755,"style":14691},[56],[17,15757,15759,15762],{"className":15758},[61],[17,15760,111],{"className":15761,"style":127},[61,62],[17,15763,15765],{"className":15764},[433],[17,15766,15768,15788],{"className":15767},[437,1313],[17,15769,15771,15785],{"className":15770},[441],[17,15772,15774],{"className":15773,"style":1404},[445],[17,15775,15776,15779],{"style":14712},[17,15777],{"className":15778,"style":453},[452],[17,15780,15782],{"className":15781},[457,458,459,460],[17,15783,3544],{"className":15784},[61,460],[17,15786,1346],{"className":15787},[1345],[17,15789,15791],{"className":15790},[441],[17,15792,15794],{"className":15793,"style":1429},[445],[17,15795],{}," contains the same terms as that of ",[17,15798,15800,15813],{"className":15799,"translate":21},[20],[17,15801,15803],{"className":15802},[25],[27,15804,15805],{"xmlns":29},[31,15806,15807,15811],{},[34,15808,15809],{},[37,15810,111],{},[41,15812,111],{"encoding":43},[17,15814,15816],{"className":15815,"ariaHidden":48},[47],[17,15817,15819,15822],{"className":15818},[52],[17,15820],{"className":15821,"style":123},[56],[17,15823,111],{"className":15824,"style":127},[61,62],", but with each term having a sign difference of ",[17,15827,15829,15907],{"className":15828,"translate":21},[20],[17,15830,15832],{"className":15831},[25],[27,15833,15834],{"xmlns":29},[31,15835,15836,15904],{},[34,15837,15838,15840,15842,15844],{},[204,15839,622],{"stretchy":621},[204,15841,627],{},[408,15843,3544],{},[403,15845,15846,15848],{},[204,15847,632],{"stretchy":621},[34,15849,15850,15856,15858,15864,15866,15868,15870,15876,15878,15884,15886,15892,15894,15896,15898],{},[848,15851,15852,15854],{},[37,15853,3890],{},[408,15855,3544],{},[204,15857,3557],{},[848,15859,15860,15862],{},[37,15861,3890],{},[408,15863,410],{},[204,15865,3557],{},[204,15867,3905],{},[204,15869,3557],{},[848,15871,15872,15874],{},[37,15873,3890],{},[37,15875,39],{},[204,15877,3557],{},[848,15879,15880,15882],{},[37,15881,3928],{},[408,15883,3544],{},[204,15885,3557],{},[848,15887,15888,15890],{},[37,15889,3928],{},[408,15891,410],{},[204,15893,3557],{},[204,15895,3905],{},[204,15897,3557],{},[848,15899,15900,15902],{},[37,15901,3928],{},[37,15903,39],{},[41,15905,15906],{"encoding":43},"(-1)^{i_1+i_2+\\cdots+i_k+j_1+j_2+\\cdots+j_k}",[17,15908,15910],{"className":15909,"ariaHidden":48},[47],[17,15911,15913,15917,15920,15923,15926],{"className":15912},[52],[17,15914],{"className":15915,"style":15916},[56],"height:1.0747em;vertical-align:-0.25em;",[17,15918,622],{"className":15919},[649],[17,15921,627],{"className":15922},[61],[17,15924,3544],{"className":15925},[61],[17,15927,15929,15932],{"className":15928},[674],[17,15930,632],{"className":15931},[674],[17,15933,15935],{"className":15934},[433],[17,15936,15938],{"className":15937},[437],[17,15939,15941],{"className":15940},[441],[17,15942,15945],{"className":15943,"style":15944},[445],"height:0.8247em;",[17,15946,15947,15950],{"style":448},[17,15948],{"className":15949,"style":453},[452],[17,15951,15953],{"className":15952},[457,458,459,460],[17,15954,15956,15996,15999,16039,16042,16045,16048,16088,16091,16131,16134,16174,16177,16180,16183],{"className":15955},[61,460],[17,15957,15959,15962],{"className":15958},[61,460],[17,15960,3890],{"className":15961},[61,62,460],[17,15963,15965],{"className":15964},[433],[17,15966,15968,15988],{"className":15967},[437,1313],[17,15969,15971,15985],{"className":15970},[441],[17,15972,15974],{"className":15973,"style":4541},[445],[17,15975,15976,15979],{"style":4544},[17,15977],{"className":15978,"style":4548},[452],[17,15980,15982],{"className":15981},[457,4552,4553,460],[17,15983,3544],{"className":15984},[61,460],[17,15986,1346],{"className":15987},[1345],[17,15989,15991],{"className":15990},[441],[17,15992,15994],{"className":15993,"style":4569},[445],[17,15995],{},[17,15997,3557],{"className":15998},[233,460],[17,16000,16002,16005],{"className":16001},[61,460],[17,16003,3890],{"className":16004},[61,62,460],[17,16006,16008],{"className":16007},[433],[17,16009,16011,16031],{"className":16010},[437,1313],[17,16012,16014,16028],{"className":16013},[441],[17,16015,16017],{"className":16016,"style":4541},[445],[17,16018,16019,16022],{"style":4544},[17,16020],{"className":16021,"style":4548},[452],[17,16023,16025],{"className":16024},[457,4552,4553,460],[17,16026,410],{"className":16027},[61,460],[17,16029,1346],{"className":16030},[1345],[17,16032,16034],{"className":16033},[441],[17,16035,16037],{"className":16036,"style":4569},[445],[17,16038],{},[17,16040,3557],{"className":16041},[233,460],[17,16043,3905],{"className":16044},[1301,460],[17,16046,3557],{"className":16047},[233,460],[17,16049,16051,16054],{"className":16050},[61,460],[17,16052,3890],{"className":16053},[61,62,460],[17,16055,16057],{"className":16056},[433],[17,16058,16060,16080],{"className":16059},[437,1313],[17,16061,16063,16077],{"className":16062},[441],[17,16064,16066],{"className":16065,"style":4645},[445],[17,16067,16068,16071],{"style":4648},[17,16069],{"className":16070,"style":4548},[452],[17,16072,16074],{"className":16073},[457,4552,4553,460],[17,16075,39],{"className":16076,"style":63},[61,62,460],[17,16078,1346],{"className":16079},[1345],[17,16081,16083],{"className":16082},[441],[17,16084,16086],{"className":16085,"style":4670},[445],[17,16087],{},[17,16089,3557],{"className":16090},[233,460],[17,16092,16094,16097],{"className":16093},[61,460],[17,16095,3928],{"className":16096,"style":4138},[61,62,460],[17,16098,16100],{"className":16099},[433],[17,16101,16103,16123],{"className":16102},[437,1313],[17,16104,16106,16120],{"className":16105},[441],[17,16107,16109],{"className":16108,"style":4541},[445],[17,16110,16111,16114],{"style":4702},[17,16112],{"className":16113,"style":4548},[452],[17,16115,16117],{"className":16116},[457,4552,4553,460],[17,16118,3544],{"className":16119},[61,460],[17,16121,1346],{"className":16122},[1345],[17,16124,16126],{"className":16125},[441],[17,16127,16129],{"className":16128,"style":4569},[445],[17,16130],{},[17,16132,3557],{"className":16133},[233,460],[17,16135,16137,16140],{"className":16136},[61,460],[17,16138,3928],{"className":16139,"style":4138},[61,62,460],[17,16141,16143],{"className":16142},[433],[17,16144,16146,16166],{"className":16145},[437,1313],[17,16147,16149,16163],{"className":16148},[441],[17,16150,16152],{"className":16151,"style":4541},[445],[17,16153,16154,16157],{"style":4702},[17,16155],{"className":16156,"style":4548},[452],[17,16158,16160],{"className":16159},[457,4552,4553,460],[17,16161,410],{"className":16162},[61,460],[17,16164,1346],{"className":16165},[1345],[17,16167,16169],{"className":16168},[441],[17,16170,16172],{"className":16171,"style":4569},[445],[17,16173],{},[17,16175,3557],{"className":16176},[233,460],[17,16178,3905],{"className":16179},[1301,460],[17,16181,3557],{"className":16182},[233,460],[17,16184,16186,16189],{"className":16185},[61,460],[17,16187,3928],{"className":16188,"style":4138},[61,62,460],[17,16190,16192],{"className":16191},[433],[17,16193,16195,16215],{"className":16194},[437,1313],[17,16196,16198,16212],{"className":16197},[441],[17,16199,16201],{"className":16200,"style":4645},[445],[17,16202,16203,16206],{"style":4801},[17,16204],{"className":16205,"style":4548},[452],[17,16207,16209],{"className":16208},[457,4552,4553,460],[17,16210,39],{"className":16211,"style":63},[61,62,460],[17,16213,1346],{"className":16214},[1345],[17,16216,16218],{"className":16217},[441],[17,16219,16221],{"className":16220,"style":4670},[445],[17,16222],{},[10,16224,16225,16226,16254,16255,16324,16325,16353,16354,16423,16424,16485,16486,16553,16554,11414],{},"Now, ",[17,16227,16229,16242],{"className":16228,"translate":21},[20],[17,16230,16232],{"className":16231},[25],[27,16233,16234],{"xmlns":29},[31,16235,16236,16240],{},[34,16237,16238],{},[37,16239,172],{},[41,16241,172],{"encoding":43},[17,16243,16245],{"className":16244,"ariaHidden":48},[47],[17,16246,16248,16251],{"className":16247},[52],[17,16249],{"className":16250,"style":123},[56],[17,16252,172],{"className":16253,"style":187},[61,62]," is located in the top-left corner of ",[17,16256,16258,16275],{"className":16257,"translate":21},[20],[17,16259,16261],{"className":16260},[25],[27,16262,16263],{"xmlns":29},[31,16264,16265,16273],{},[34,16266,16267],{},[848,16268,16269,16271],{},[37,16270,111],{},[408,16272,3544],{},[41,16274,14681],{"encoding":43},[17,16276,16278],{"className":16277,"ariaHidden":48},[47],[17,16279,16281,16284],{"className":16280},[52],[17,16282],{"className":16283,"style":14691},[56],[17,16285,16287,16290],{"className":16286},[61],[17,16288,111],{"className":16289,"style":127},[61,62],[17,16291,16293],{"className":16292},[433],[17,16294,16296,16316],{"className":16295},[437,1313],[17,16297,16299,16313],{"className":16298},[441],[17,16300,16302],{"className":16301,"style":1404},[445],[17,16303,16304,16307],{"style":14712},[17,16305],{"className":16306,"style":453},[452],[17,16308,16310],{"className":16309},[457,458,459,460],[17,16311,3544],{"className":16312},[61,460],[17,16314,1346],{"className":16315},[1345],[17,16317,16319],{"className":16318},[441],[17,16320,16322],{"className":16321,"style":1429},[445],[17,16323],{},", and the cofactor and algebraic cofactor of ",[17,16326,16328,16341],{"className":16327,"translate":21},[20],[17,16329,16331],{"className":16330},[25],[27,16332,16333],{"xmlns":29},[31,16334,16335,16339],{},[34,16336,16337],{},[37,16338,172],{},[41,16340,172],{"encoding":43},[17,16342,16344],{"className":16343,"ariaHidden":48},[47],[17,16345,16347,16350],{"className":16346},[52],[17,16348],{"className":16349,"style":123},[56],[17,16351,172],{"className":16352,"style":187},[61,62]," in ",[17,16355,16357,16374],{"className":16356,"translate":21},[20],[17,16358,16360],{"className":16359},[25],[27,16361,16362],{"xmlns":29},[31,16363,16364,16372],{},[34,16365,16366],{},[848,16367,16368,16370],{},[37,16369,111],{},[408,16371,3544],{},[41,16373,14681],{"encoding":43},[17,16375,16377],{"className":16376,"ariaHidden":48},[47],[17,16378,16380,16383],{"className":16379},[52],[17,16381],{"className":16382,"style":14691},[56],[17,16384,16386,16389],{"className":16385},[61],[17,16387,111],{"className":16388,"style":127},[61,62],[17,16390,16392],{"className":16391},[433],[17,16393,16395,16415],{"className":16394},[437,1313],[17,16396,16398,16412],{"className":16397},[441],[17,16399,16401],{"className":16400,"style":1404},[445],[17,16402,16403,16406],{"style":14712},[17,16404],{"className":16405,"style":453},[452],[17,16407,16409],{"className":16408},[457,458,459,460],[17,16410,3544],{"className":16411},[61,460],[17,16413,1346],{"className":16414},[1345],[17,16416,16418],{"className":16417},[441],[17,16419,16421],{"className":16420,"style":1429},[445],[17,16422],{}," are both ",[17,16425,16427,16444],{"className":16426,"translate":21},[20],[17,16428,16430],{"className":16429},[25],[27,16431,16432],{"xmlns":29},[31,16433,16434,16442],{},[34,16435,16436],{},[403,16437,16438,16440],{},[37,16439,172],{},[204,16441,695],{"mathvariant":693,"lspace":694,"rspace":694},[41,16443,698],{"encoding":43},[17,16445,16447],{"className":16446,"ariaHidden":48},[47],[17,16448,16450,16453],{"className":16449},[52],[17,16451],{"className":16452,"style":708},[56],[17,16454,16456,16459],{"className":16455},[61],[17,16457,172],{"className":16458,"style":187},[61,62],[17,16460,16462],{"className":16461},[433],[17,16463,16465],{"className":16464},[437],[17,16466,16468],{"className":16467},[441],[17,16469,16471],{"className":16470,"style":708},[445],[17,16472,16473,16476],{"style":448},[17,16474],{"className":16475,"style":453},[452],[17,16477,16479],{"className":16478},[457,458,459,460],[17,16480,16482],{"className":16481},[61,460],[17,16483,695],{"className":16484},[61,460],". Therefore, each term in the product ",[17,16487,16489,16509],{"className":16488,"translate":21},[20],[17,16490,16492],{"className":16491},[25],[27,16493,16494],{"xmlns":29},[31,16495,16496,16506],{},[34,16497,16498,16500],{},[37,16499,172],{},[403,16501,16502,16504],{},[37,16503,172],{},[204,16505,695],{"mathvariant":693,"lspace":694,"rspace":694},[41,16507,16508],{"encoding":43},"MM'",[17,16510,16512],{"className":16511,"ariaHidden":48},[47],[17,16513,16515,16518,16521],{"className":16514},[52],[17,16516],{"className":16517,"style":708},[56],[17,16519,172],{"className":16520,"style":187},[61,62],[17,16522,16524,16527],{"className":16523},[61],[17,16525,172],{"className":16526,"style":187},[61,62],[17,16528,16530],{"className":16529},[433],[17,16531,16533],{"className":16532},[437],[17,16534,16536],{"className":16535},[441],[17,16537,16539],{"className":16538,"style":708},[445],[17,16540,16541,16544],{"style":448},[17,16542],{"className":16543,"style":453},[452],[17,16545,16547],{"className":16546},[457,458,459,460],[17,16548,16550],{"className":16549},[61,460],[17,16551,695],{"className":16552},[61,460]," corresponds to a term in ",[17,16555,16557,16574],{"className":16556,"translate":21},[20],[17,16558,16560],{"className":16559},[25],[27,16561,16562],{"xmlns":29},[31,16563,16564,16572],{},[34,16565,16566],{},[848,16567,16568,16570],{},[37,16569,111],{},[408,16571,3544],{},[41,16573,14681],{"encoding":43},[17,16575,16577],{"className":16576,"ariaHidden":48},[47],[17,16578,16580,16583],{"className":16579},[52],[17,16581],{"className":16582,"style":14691},[56],[17,16584,16586,16589],{"className":16585},[61],[17,16587,111],{"className":16588,"style":127},[61,62],[17,16590,16592],{"className":16591},[433],[17,16593,16595,16615],{"className":16594},[437,1313],[17,16596,16598,16612],{"className":16597},[441],[17,16599,16601],{"className":16600,"style":1404},[445],[17,16602,16603,16606],{"style":14712},[17,16604],{"className":16605,"style":453},[452],[17,16607,16609],{"className":16608},[457,458,459,460],[17,16610,3544],{"className":16611},[61,460],[17,16613,1346],{"className":16614},[1345],[17,16616,16618],{"className":16617},[441],[17,16619,16621],{"className":16620,"style":1429},[445],[17,16622],{},[10,16624,16625,16626,4352,16654,16353,16682,16710,16711,17176,17177,17246,17247,17641,17642,17676,17677,11414],{},"The algebraic cofactor ",[17,16627,16629,16642],{"className":16628,"translate":21},[20],[17,16630,16632],{"className":16631},[25],[27,16633,16634],{"xmlns":29},[31,16635,16636,16640],{},[34,16637,16638],{},[37,16639,4935],{},[41,16641,4935],{"encoding":43},[17,16643,16645],{"className":16644,"ariaHidden":48},[47],[17,16646,16648,16651],{"className":16647},[52],[17,16649],{"className":16650,"style":123},[56],[17,16652,4935],{"className":16653},[61,62],[17,16655,16657,16670],{"className":16656,"translate":21},[20],[17,16658,16660],{"className":16659},[25],[27,16661,16662],{"xmlns":29},[31,16663,16664,16668],{},[34,16665,16666],{},[37,16667,172],{},[41,16669,172],{"encoding":43},[17,16671,16673],{"className":16672,"ariaHidden":48},[47],[17,16674,16676,16679],{"className":16675},[52],[17,16677],{"className":16678,"style":123},[56],[17,16680,172],{"className":16681,"style":187},[61,62],[17,16683,16685,16698],{"className":16684,"translate":21},[20],[17,16686,16688],{"className":16687},[25],[27,16689,16690],{"xmlns":29},[31,16691,16692,16696],{},[34,16693,16694],{},[37,16695,111],{},[41,16697,111],{"encoding":43},[17,16699,16701],{"className":16700,"ariaHidden":48},[47],[17,16702,16704,16707],{"className":16703},[52],[17,16705],{"className":16706,"style":123},[56],[17,16708,111],{"className":16709,"style":127},[61,62]," gives ",[17,16712,16714,16806],{"className":16713,"translate":21},[20],[17,16715,16717],{"className":16716},[25],[27,16718,16719],{"xmlns":29},[31,16720,16721,16803],{},[34,16722,16723,16725,16727,16729,16731,16733,16735,16795,16797],{},[37,16724,172],{},[37,16726,4935],{},[204,16728,825],{},[204,16730,622],{"stretchy":621},[204,16732,627],{},[408,16734,3544],{},[403,16736,16737,16739],{},[204,16738,632],{"stretchy":621},[34,16740,16741,16747,16749,16755,16757,16759,16761,16767,16769,16775,16777,16783,16785,16787,16789],{},[848,16742,16743,16745],{},[37,16744,3890],{},[408,16746,3544],{},[204,16748,3557],{},[848,16750,16751,16753],{},[37,16752,3890],{},[408,16754,410],{},[204,16756,3557],{},[204,16758,3905],{},[204,16760,3557],{},[848,16762,16763,16765],{},[37,16764,3890],{},[37,16766,39],{},[204,16768,3557],{},[848,16770,16771,16773],{},[37,16772,3928],{},[408,16774,3544],{},[204,16776,3557],{},[848,16778,16779,16781],{},[37,16780,3928],{},[408,16782,410],{},[204,16784,3557],{},[204,16786,3905],{},[204,16788,3557],{},[848,16790,16791,16793],{},[37,16792,3928],{},[37,16794,39],{},[37,16796,172],{},[403,16798,16799,16801],{},[37,16800,172],{},[204,16802,695],{"mathvariant":693,"lspace":694,"rspace":694},[41,16804,16805],{"encoding":43},"MA = (-1)^{i_1+i_2+\\cdots+i_k+j_1+j_2+\\cdots+j_k}MM'",[17,16807,16809,16830],{"className":16808,"ariaHidden":48},[47],[17,16810,16812,16815,16818,16821,16824,16827],{"className":16811},[52],[17,16813],{"className":16814,"style":123},[56],[17,16816,172],{"className":16817,"style":187},[61,62],[17,16819,4935],{"className":16820},[61,62],[17,16822],{"className":16823,"style":372},[228],[17,16825,825],{"className":16826},[376],[17,16828],{"className":16829,"style":372},[228],[17,16831,16833,16836,16839,16842,16845,17141,17144],{"className":16832},[52],[17,16834],{"className":16835,"style":15916},[56],[17,16837,622],{"className":16838},[649],[17,16840,627],{"className":16841},[61],[17,16843,3544],{"className":16844},[61],[17,16846,16848,16851],{"className":16847},[674],[17,16849,632],{"className":16850},[674],[17,16852,16854],{"className":16853},[433],[17,16855,16857],{"className":16856},[437],[17,16858,16860],{"className":16859},[441],[17,16861,16863],{"className":16862,"style":15944},[445],[17,16864,16865,16868],{"style":448},[17,16866],{"className":16867,"style":453},[452],[17,16869,16871],{"className":16870},[457,458,459,460],[17,16872,16874,16914,16917,16957,16960,16963,16966,17006,17009,17049,17052,17092,17095,17098,17101],{"className":16873},[61,460],[17,16875,16877,16880],{"className":16876},[61,460],[17,16878,3890],{"className":16879},[61,62,460],[17,16881,16883],{"className":16882},[433],[17,16884,16886,16906],{"className":16885},[437,1313],[17,16887,16889,16903],{"className":16888},[441],[17,16890,16892],{"className":16891,"style":4541},[445],[17,16893,16894,16897],{"style":4544},[17,16895],{"className":16896,"style":4548},[452],[17,16898,16900],{"className":16899},[457,4552,4553,460],[17,16901,3544],{"className":16902},[61,460],[17,16904,1346],{"className":16905},[1345],[17,16907,16909],{"className":16908},[441],[17,16910,16912],{"className":16911,"style":4569},[445],[17,16913],{},[17,16915,3557],{"className":16916},[233,460],[17,16918,16920,16923],{"className":16919},[61,460],[17,16921,3890],{"className":16922},[61,62,460],[17,16924,16926],{"className":16925},[433],[17,16927,16929,16949],{"className":16928},[437,1313],[17,16930,16932,16946],{"className":16931},[441],[17,16933,16935],{"className":16934,"style":4541},[445],[17,16936,16937,16940],{"style":4544},[17,16938],{"className":16939,"style":4548},[452],[17,16941,16943],{"className":16942},[457,4552,4553,460],[17,16944,410],{"className":16945},[61,460],[17,16947,1346],{"className":16948},[1345],[17,16950,16952],{"className":16951},[441],[17,16953,16955],{"className":16954,"style":4569},[445],[17,16956],{},[17,16958,3557],{"className":16959},[233,460],[17,16961,3905],{"className":16962},[1301,460],[17,16964,3557],{"className":16965},[233,460],[17,16967,16969,16972],{"className":16968},[61,460],[17,16970,3890],{"className":16971},[61,62,460],[17,16973,16975],{"className":16974},[433],[17,16976,16978,16998],{"className":16977},[437,1313],[17,16979,16981,16995],{"className":16980},[441],[17,16982,16984],{"className":16983,"style":4645},[445],[17,16985,16986,16989],{"style":4648},[17,16987],{"className":16988,"style":4548},[452],[17,16990,16992],{"className":16991},[457,4552,4553,460],[17,16993,39],{"className":16994,"style":63},[61,62,460],[17,16996,1346],{"className":16997},[1345],[17,16999,17001],{"className":17000},[441],[17,17002,17004],{"className":17003,"style":4670},[445],[17,17005],{},[17,17007,3557],{"className":17008},[233,460],[17,17010,17012,17015],{"className":17011},[61,460],[17,17013,3928],{"className":17014,"style":4138},[61,62,460],[17,17016,17018],{"className":17017},[433],[17,17019,17021,17041],{"className":17020},[437,1313],[17,17022,17024,17038],{"className":17023},[441],[17,17025,17027],{"className":17026,"style":4541},[445],[17,17028,17029,17032],{"style":4702},[17,17030],{"className":17031,"style":4548},[452],[17,17033,17035],{"className":17034},[457,4552,4553,460],[17,17036,3544],{"className":17037},[61,460],[17,17039,1346],{"className":17040},[1345],[17,17042,17044],{"className":17043},[441],[17,17045,17047],{"className":17046,"style":4569},[445],[17,17048],{},[17,17050,3557],{"className":17051},[233,460],[17,17053,17055,17058],{"className":17054},[61,460],[17,17056,3928],{"className":17057,"style":4138},[61,62,460],[17,17059,17061],{"className":17060},[433],[17,17062,17064,17084],{"className":17063},[437,1313],[17,17065,17067,17081],{"className":17066},[441],[17,17068,17070],{"className":17069,"style":4541},[445],[17,17071,17072,17075],{"style":4702},[17,17073],{"className":17074,"style":4548},[452],[17,17076,17078],{"className":17077},[457,4552,4553,460],[17,17079,410],{"className":17080},[61,460],[17,17082,1346],{"className":17083},[1345],[17,17085,17087],{"className":17086},[441],[17,17088,17090],{"className":17089,"style":4569},[445],[17,17091],{},[17,17093,3557],{"className":17094},[233,460],[17,17096,3905],{"className":17097},[1301,460],[17,17099,3557],{"className":17100},[233,460],[17,17102,17104,17107],{"className":17103},[61,460],[17,17105,3928],{"className":17106,"style":4138},[61,62,460],[17,17108,17110],{"className":17109},[433],[17,17111,17113,17133],{"className":17112},[437,1313],[17,17114,17116,17130],{"className":17115},[441],[17,17117,17119],{"className":17118,"style":4645},[445],[17,17120,17121,17124],{"style":4801},[17,17122],{"className":17123,"style":4548},[452],[17,17125,17127],{"className":17126},[457,4552,4553,460],[17,17128,39],{"className":17129,"style":63},[61,62,460],[17,17131,1346],{"className":17132},[1345],[17,17134,17136],{"className":17135},[441],[17,17137,17139],{"className":17138,"style":4670},[445],[17,17140],{},[17,17142,172],{"className":17143,"style":187},[61,62],[17,17145,17147,17150],{"className":17146},[61],[17,17148,172],{"className":17149,"style":187},[61,62],[17,17151,17153],{"className":17152},[433],[17,17154,17156],{"className":17155},[437],[17,17157,17159],{"className":17158},[441],[17,17160,17162],{"className":17161,"style":708},[445],[17,17163,17164,17167],{"style":448},[17,17165],{"className":17166,"style":453},[452],[17,17168,17170],{"className":17169},[457,458,459,460],[17,17171,17173],{"className":17172},[61,460],[17,17174,695],{"className":17175},[61,460],", which is a term in ",[17,17178,17180,17197],{"className":17179,"translate":21},[20],[17,17181,17183],{"className":17182},[25],[27,17184,17185],{"xmlns":29},[31,17186,17187,17195],{},[34,17188,17189],{},[848,17190,17191,17193],{},[37,17192,111],{},[408,17194,3544],{},[41,17196,14681],{"encoding":43},[17,17198,17200],{"className":17199,"ariaHidden":48},[47],[17,17201,17203,17206],{"className":17202},[52],[17,17204],{"className":17205,"style":14691},[56],[17,17207,17209,17212],{"className":17208},[61],[17,17210,111],{"className":17211,"style":127},[61,62],[17,17213,17215],{"className":17214},[433],[17,17216,17218,17238],{"className":17217},[437,1313],[17,17219,17221,17235],{"className":17220},[441],[17,17222,17224],{"className":17223,"style":1404},[445],[17,17225,17226,17229],{"style":14712},[17,17227],{"className":17228,"style":453},[452],[17,17230,17232],{"className":17231},[457,458,459,460],[17,17233,3544],{"className":17234},[61,460],[17,17236,1346],{"className":17237},[1345],[17,17239,17241],{"className":17240},[441],[17,17242,17244],{"className":17243,"style":1429},[445],[17,17245],{}," with a sign difference of ",[17,17248,17250,17327],{"className":17249,"translate":21},[20],[17,17251,17253],{"className":17252},[25],[27,17254,17255],{"xmlns":29},[31,17256,17257,17325],{},[34,17258,17259,17261,17263,17265],{},[204,17260,622],{"stretchy":621},[204,17262,627],{},[408,17264,3544],{},[403,17266,17267,17269],{},[204,17268,632],{"stretchy":621},[34,17270,17271,17277,17279,17285,17287,17289,17291,17297,17299,17305,17307,17313,17315,17317,17319],{},[848,17272,17273,17275],{},[37,17274,3890],{},[408,17276,3544],{},[204,17278,3557],{},[848,17280,17281,17283],{},[37,17282,3890],{},[408,17284,410],{},[204,17286,3557],{},[204,17288,3905],{},[204,17290,3557],{},[848,17292,17293,17295],{},[37,17294,3890],{},[37,17296,39],{},[204,17298,3557],{},[848,17300,17301,17303],{},[37,17302,3928],{},[408,17304,3544],{},[204,17306,3557],{},[848,17308,17309,17311],{},[37,17310,3928],{},[408,17312,410],{},[204,17314,3557],{},[204,17316,3905],{},[204,17318,3557],{},[848,17320,17321,17323],{},[37,17322,3928],{},[37,17324,39],{},[41,17326,15906],{"encoding":43},[17,17328,17330],{"className":17329,"ariaHidden":48},[47],[17,17331,17333,17336,17339,17342,17345],{"className":17332},[52],[17,17334],{"className":17335,"style":15916},[56],[17,17337,622],{"className":17338},[649],[17,17340,627],{"className":17341},[61],[17,17343,3544],{"className":17344},[61],[17,17346,17348,17351],{"className":17347},[674],[17,17349,632],{"className":17350},[674],[17,17352,17354],{"className":17353},[433],[17,17355,17357],{"className":17356},[437],[17,17358,17360],{"className":17359},[441],[17,17361,17363],{"className":17362,"style":15944},[445],[17,17364,17365,17368],{"style":448},[17,17366],{"className":17367,"style":453},[452],[17,17369,17371],{"className":17370},[457,458,459,460],[17,17372,17374,17414,17417,17457,17460,17463,17466,17506,17509,17549,17552,17592,17595,17598,17601],{"className":17373},[61,460],[17,17375,17377,17380],{"className":17376},[61,460],[17,17378,3890],{"className":17379},[61,62,460],[17,17381,17383],{"className":17382},[433],[17,17384,17386,17406],{"className":17385},[437,1313],[17,17387,17389,17403],{"className":17388},[441],[17,17390,17392],{"className":17391,"style":4541},[445],[17,17393,17394,17397],{"style":4544},[17,17395],{"className":17396,"style":4548},[452],[17,17398,17400],{"className":17399},[457,4552,4553,460],[17,17401,3544],{"className":17402},[61,460],[17,17404,1346],{"className":17405},[1345],[17,17407,17409],{"className":17408},[441],[17,17410,17412],{"className":17411,"style":4569},[445],[17,17413],{},[17,17415,3557],{"className":17416},[233,460],[17,17418,17420,17423],{"className":17419},[61,460],[17,17421,3890],{"className":17422},[61,62,460],[17,17424,17426],{"className":17425},[433],[17,17427,17429,17449],{"className":17428},[437,1313],[17,17430,17432,17446],{"className":17431},[441],[17,17433,17435],{"className":17434,"style":4541},[445],[17,17436,17437,17440],{"style":4544},[17,17438],{"className":17439,"style":4548},[452],[17,17441,17443],{"className":17442},[457,4552,4553,460],[17,17444,410],{"className":17445},[61,460],[17,17447,1346],{"className":17448},[1345],[17,17450,17452],{"className":17451},[441],[17,17453,17455],{"className":17454,"style":4569},[445],[17,17456],{},[17,17458,3557],{"className":17459},[233,460],[17,17461,3905],{"className":17462},[1301,460],[17,17464,3557],{"className":17465},[233,460],[17,17467,17469,17472],{"className":17468},[61,460],[17,17470,3890],{"className":17471},[61,62,460],[17,17473,17475],{"className":17474},[433],[17,17476,17478,17498],{"className":17477},[437,1313],[17,17479,17481,17495],{"className":17480},[441],[17,17482,17484],{"className":17483,"style":4645},[445],[17,17485,17486,17489],{"style":4648},[17,17487],{"className":17488,"style":4548},[452],[17,17490,17492],{"className":17491},[457,4552,4553,460],[17,17493,39],{"className":17494,"style":63},[61,62,460],[17,17496,1346],{"className":17497},[1345],[17,17499,17501],{"className":17500},[441],[17,17502,17504],{"className":17503,"style":4670},[445],[17,17505],{},[17,17507,3557],{"className":17508},[233,460],[17,17510,17512,17515],{"className":17511},[61,460],[17,17513,3928],{"className":17514,"style":4138},[61,62,460],[17,17516,17518],{"className":17517},[433],[17,17519,17521,17541],{"className":17520},[437,1313],[17,17522,17524,17538],{"className":17523},[441],[17,17525,17527],{"className":17526,"style":4541},[445],[17,17528,17529,17532],{"style":4702},[17,17530],{"className":17531,"style":4548},[452],[17,17533,17535],{"className":17534},[457,4552,4553,460],[17,17536,3544],{"className":17537},[61,460],[17,17539,1346],{"className":17540},[1345],[17,17542,17544],{"className":17543},[441],[17,17545,17547],{"className":17546,"style":4569},[445],[17,17548],{},[17,17550,3557],{"className":17551},[233,460],[17,17553,17555,17558],{"className":17554},[61,460],[17,17556,3928],{"className":17557,"style":4138},[61,62,460],[17,17559,17561],{"className":17560},[433],[17,17562,17564,17584],{"className":17563},[437,1313],[17,17565,17567,17581],{"className":17566},[441],[17,17568,17570],{"className":17569,"style":4541},[445],[17,17571,17572,17575],{"style":4702},[17,17573],{"className":17574,"style":4548},[452],[17,17576,17578],{"className":17577},[457,4552,4553,460],[17,17579,410],{"className":17580},[61,460],[17,17582,1346],{"className":17583},[1345],[17,17585,17587],{"className":17586},[441],[17,17588,17590],{"className":17589,"style":4569},[445],[17,17591],{},[17,17593,3557],{"className":17594},[233,460],[17,17596,3905],{"className":17597},[1301,460],[17,17599,3557],{"className":17600},[233,460],[17,17602,17604,17607],{"className":17603},[61,460],[17,17605,3928],{"className":17606,"style":4138},[61,62,460],[17,17608,17610],{"className":17609},[433],[17,17611,17613,17633],{"className":17612},[437,1313],[17,17614,17616,17630],{"className":17615},[441],[17,17617,17619],{"className":17618,"style":4645},[445],[17,17620,17621,17624],{"style":4801},[17,17622],{"className":17623,"style":4548},[452],[17,17625,17627],{"className":17626},[457,4552,4553,460],[17,17628,39],{"className":17629,"style":63},[61,62,460],[17,17631,1346],{"className":17632},[1345],[17,17634,17636],{"className":17635},[441],[17,17637,17639],{"className":17638,"style":4670},[445],[17,17640],{},". Hence, ",[17,17643,17645,17661],{"className":17644,"translate":21},[20],[17,17646,17648],{"className":17647},[25],[27,17649,17650],{"xmlns":29},[31,17651,17652,17658],{},[34,17653,17654,17656],{},[37,17655,172],{},[37,17657,4935],{},[41,17659,17660],{"encoding":43},"MA",[17,17662,17664],{"className":17663,"ariaHidden":48},[47],[17,17665,17667,17670,17673],{"className":17666},[52],[17,17668],{"className":17669,"style":123},[56],[17,17671,172],{"className":17672,"style":187},[61,62],[17,17674,4935],{"className":17675},[61,62]," is a term in ",[17,17678,17680,17693],{"className":17679,"translate":21},[20],[17,17681,17683],{"className":17682},[25],[27,17684,17685],{"xmlns":29},[31,17686,17687,17691],{},[34,17688,17689],{},[37,17690,111],{},[41,17692,111],{"encoding":43},[17,17694,17696],{"className":17695,"ariaHidden":48},[47],[17,17697,17699,17702],{"className":17698},[52],[17,17700],{"className":17701,"style":123},[56],[17,17703,111],{"className":17704,"style":127},[61,62],[10,17706,17707,17710,17711,333,17739,17835,17836,17864,17865,17893,17894,798],{},[13,17708,17709],{},"Theorem (Laplace's Theorem):"," For any ",[17,17712,17714,17727],{"className":17713,"translate":21},[20],[17,17715,17717],{"className":17716},[25],[27,17718,17719],{"xmlns":29},[31,17720,17721,17725],{},[34,17722,17723],{},[37,17724,39],{},[41,17726,39],{"encoding":43},[17,17728,17730],{"className":17729,"ariaHidden":48},[47],[17,17731,17733,17736],{"className":17732},[52],[17,17734],{"className":17735,"style":57},[56],[17,17737,39],{"className":17738,"style":63},[61,62],[17,17740,17742,17768],{"className":17741,"translate":21},[20],[17,17743,17745],{"className":17744},[25],[27,17746,17747],{"xmlns":29},[31,17748,17749,17765],{},[34,17750,17751,17753,17755,17757,17759,17761,17763],{},[408,17752,3544],{},[204,17754,350],{},[37,17756,39],{},[204,17758,350],{},[37,17760,80],{},[204,17762,627],{},[408,17764,3544],{},[41,17766,17767],{"encoding":43},"1 \\le k \\le n-1",[17,17769,17771,17790,17808,17826],{"className":17770,"ariaHidden":48},[47],[17,17772,17774,17778,17781,17784,17787],{"className":17773},[52],[17,17775],{"className":17776,"style":17777},[56],"height:0.7804em;vertical-align:-0.136em;",[17,17779,3544],{"className":17780},[61],[17,17782],{"className":17783,"style":372},[228],[17,17785,350],{"className":17786},[376],[17,17788],{"className":17789,"style":372},[228],[17,17791,17793,17796,17799,17802,17805],{"className":17792},[52],[17,17794],{"className":17795,"style":365},[56],[17,17797,39],{"className":17798,"style":63},[61,62],[17,17800],{"className":17801,"style":372},[228],[17,17803,350],{"className":17804},[376],[17,17806],{"className":17807,"style":372},[228],[17,17809,17811,17814,17817,17820,17823],{"className":17810},[52],[17,17812],{"className":17813,"style":13282},[56],[17,17815,80],{"className":17816},[61,62],[17,17818],{"className":17819,"style":229},[228],[17,17821,627],{"className":17822},[233],[17,17824],{"className":17825,"style":229},[228],[17,17827,17829,17832],{"className":17828},[52],[17,17830],{"className":17831,"style":12756},[56],[17,17833,3544],{"className":17834},[61],") rows chosen from determinant ",[17,17837,17839,17852],{"className":17838,"translate":21},[20],[17,17840,17842],{"className":17841},[25],[27,17843,17844],{"xmlns":29},[31,17845,17846,17850],{},[34,17847,17848],{},[37,17849,111],{},[41,17851,111],{"encoding":43},[17,17853,17855],{"className":17854,"ariaHidden":48},[47],[17,17856,17858,17861],{"className":17857},[52],[17,17859],{"className":17860,"style":123},[56],[17,17862,111],{"className":17863,"style":127},[61,62],", the sum of the products of all ",[17,17866,17868,17881],{"className":17867,"translate":21},[20],[17,17869,17871],{"className":17870},[25],[27,17872,17873],{"xmlns":29},[31,17874,17875,17879],{},[34,17876,17877],{},[37,17878,39],{},[41,17880,39],{"encoding":43},[17,17882,17884],{"className":17883,"ariaHidden":48},[47],[17,17885,17887,17890],{"className":17886},[52],[17,17888],{"className":17889,"style":57},[56],[17,17891,39],{"className":17892,"style":63},[61,62],"-th order minors formed from these rows and their respective algebraic cofactors equals the determinant ",[17,17895,17897,17910],{"className":17896,"translate":21},[20],[17,17898,17900],{"className":17899},[25],[27,17901,17902],{"xmlns":29},[31,17903,17904,17908],{},[34,17905,17906],{},[37,17907,111],{},[41,17909,111],{"encoding":43},[17,17911,17913],{"className":17912,"ariaHidden":48},[47],[17,17914,17916,17919],{"className":17915},[52],[17,17917],{"className":17918,"style":123},[56],[17,17920,111],{"className":17921,"style":127},[61,62],[10,17923,17924,17926,17927,17955,17956,17984,17985,18188,18189,18388],{},[13,17925,4984],{}," Let the minors formed from the chosen ",[17,17928,17930,17943],{"className":17929,"translate":21},[20],[17,17931,17933],{"className":17932},[25],[27,17934,17935],{"xmlns":29},[31,17936,17937,17941],{},[34,17938,17939],{},[37,17940,39],{},[41,17942,39],{"encoding":43},[17,17944,17946],{"className":17945,"ariaHidden":48},[47],[17,17947,17949,17952],{"className":17948},[52],[17,17950],{"className":17951,"style":57},[56],[17,17953,39],{"className":17954,"style":63},[61,62]," rows of ",[17,17957,17959,17972],{"className":17958,"translate":21},[20],[17,17960,17962],{"className":17961},[25],[27,17963,17964],{"xmlns":29},[31,17965,17966,17970],{},[34,17967,17968],{},[37,17969,111],{},[41,17971,111],{"encoding":43},[17,17973,17975],{"className":17974,"ariaHidden":48},[47],[17,17976,17978,17981],{"className":17977},[52],[17,17979],{"className":17980,"style":123},[56],[17,17982,111],{"className":17983,"style":127},[61,62]," be ",[17,17986,17988,18029],{"className":17987,"translate":21},[20],[17,17989,17991],{"className":17990},[25],[27,17992,17993],{"xmlns":29},[31,17994,17995,18026],{},[34,17996,17997,18003,18005,18011,18013,18015,18017,18019],{},[848,17998,17999,18001],{},[37,18000,172],{},[408,18002,3544],{},[204,18004,1084],{"separator":48},[848,18006,18007,18009],{},[37,18008,172],{},[408,18010,410],{},[204,18012,1084],{"separator":48},[204,18014,3905],{},[3907,18016,3909],{},[204,18018,1084],{"separator":48},[848,18020,18021,18023],{},[37,18022,172],{},[37,18024,18025],{},"t",[41,18027,18028],{"encoding":43},"M_1, M_2, \\cdots, M_t",[17,18030,18032],{"className":18031,"ariaHidden":48},[47],[17,18033,18035,18039,18080,18083,18086,18126,18129,18132,18135,18138,18141,18144,18147],{"className":18034},[52],[17,18036],{"className":18037,"style":18038},[56],"height:0.8778em;vertical-align:-0.1944em;",[17,18040,18042,18045],{"className":18041},[61],[17,18043,172],{"className":18044,"style":187},[61,62],[17,18046,18048],{"className":18047},[433],[17,18049,18051,18072],{"className":18050},[437,1313],[17,18052,18054,18069],{"className":18053},[441],[17,18055,18057],{"className":18056,"style":1404},[445],[17,18058,18060,18063],{"style":18059},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[17,18061],{"className":18062,"style":453},[452],[17,18064,18066],{"className":18065},[457,458,459,460],[17,18067,3544],{"className":18068},[61,460],[17,18070,1346],{"className":18071},[1345],[17,18073,18075],{"className":18074},[441],[17,18076,18078],{"className":18077,"style":1429},[445],[17,18079],{},[17,18081,1084],{"className":18082},[2410],[17,18084],{"className":18085,"style":2406},[228],[17,18087,18089,18092],{"className":18088},[61],[17,18090,172],{"className":18091,"style":187},[61,62],[17,18093,18095],{"className":18094},[433],[17,18096,18098,18118],{"className":18097},[437,1313],[17,18099,18101,18115],{"className":18100},[441],[17,18102,18104],{"className":18103,"style":1404},[445],[17,18105,18106,18109],{"style":18059},[17,18107],{"className":18108,"style":453},[452],[17,18110,18112],{"className":18111},[457,458,459,460],[17,18113,410],{"className":18114},[61,460],[17,18116,1346],{"className":18117},[1345],[17,18119,18121],{"className":18120},[441],[17,18122,18124],{"className":18123,"style":1429},[445],[17,18125],{},[17,18127,1084],{"className":18128},[2410],[17,18130],{"className":18131,"style":2406},[228],[17,18133,3905],{"className":18134},[1301],[17,18136],{"className":18137,"style":2406},[228],[17,18139],{"className":18140,"style":2406},[228],[17,18142,1084],{"className":18143},[2410],[17,18145],{"className":18146,"style":2406},[228],[17,18148,18150,18153],{"className":18149},[61],[17,18151,172],{"className":18152,"style":187},[61,62],[17,18154,18156],{"className":18155},[433],[17,18157,18159,18180],{"className":18158},[437,1313],[17,18160,18162,18177],{"className":18161},[441],[17,18163,18166],{"className":18164,"style":18165},[445],"height:0.2806em;",[17,18167,18168,18171],{"style":18059},[17,18169],{"className":18170,"style":453},[452],[17,18172,18174],{"className":18173},[457,458,459,460],[17,18175,18025],{"className":18176},[61,62,460],[17,18178,1346],{"className":18179},[1345],[17,18181,18183],{"className":18182},[441],[17,18184,18186],{"className":18185,"style":1429},[445],[17,18187],{},", with algebraic cofactors ",[17,18190,18192,18232],{"className":18191,"translate":21},[20],[17,18193,18195],{"className":18194},[25],[27,18196,18197],{"xmlns":29},[31,18198,18199,18229],{},[34,18200,18201,18207,18209,18215,18217,18219,18221,18223],{},[848,18202,18203,18205],{},[37,18204,4935],{},[408,18206,3544],{},[204,18208,1084],{"separator":48},[848,18210,18211,18213],{},[37,18212,4935],{},[408,18214,410],{},[204,18216,1084],{"separator":48},[204,18218,3905],{},[3907,18220,3909],{},[204,18222,1084],{"separator":48},[848,18224,18225,18227],{},[37,18226,4935],{},[37,18228,18025],{},[41,18230,18231],{"encoding":43},"A_1, A_2, \\cdots, A_t",[17,18233,18235],{"className":18234,"ariaHidden":48},[47],[17,18236,18238,18241,18281,18284,18287,18327,18330,18333,18336,18339,18342,18345,18348],{"className":18237},[52],[17,18239],{"className":18240,"style":18038},[56],[17,18242,18244,18247],{"className":18243},[61],[17,18245,4935],{"className":18246},[61,62],[17,18248,18250],{"className":18249},[433],[17,18251,18253,18273],{"className":18252},[437,1313],[17,18254,18256,18270],{"className":18255},[441],[17,18257,18259],{"className":18258,"style":1404},[445],[17,18260,18261,18264],{"style":1407},[17,18262],{"className":18263,"style":453},[452],[17,18265,18267],{"className":18266},[457,458,459,460],[17,18268,3544],{"className":18269},[61,460],[17,18271,1346],{"className":18272},[1345],[17,18274,18276],{"className":18275},[441],[17,18277,18279],{"className":18278,"style":1429},[445],[17,18280],{},[17,18282,1084],{"className":18283},[2410],[17,18285],{"className":18286,"style":2406},[228],[17,18288,18290,18293],{"className":18289},[61],[17,18291,4935],{"className":18292},[61,62],[17,18294,18296],{"className":18295},[433],[17,18297,18299,18319],{"className":18298},[437,1313],[17,18300,18302,18316],{"className":18301},[441],[17,18303,18305],{"className":18304,"style":1404},[445],[17,18306,18307,18310],{"style":1407},[17,18308],{"className":18309,"style":453},[452],[17,18311,18313],{"className":18312},[457,458,459,460],[17,18314,410],{"className":18315},[61,460],[17,18317,1346],{"className":18318},[1345],[17,18320,18322],{"className":18321},[441],[17,18323,18325],{"className":18324,"style":1429},[445],[17,18326],{},[17,18328,1084],{"className":18329},[2410],[17,18331],{"className":18332,"style":2406},[228],[17,18334,3905],{"className":18335},[1301],[17,18337],{"className":18338,"style":2406},[228],[17,18340],{"className":18341,"style":2406},[228],[17,18343,1084],{"className":18344},[2410],[17,18346],{"className":18347,"style":2406},[228],[17,18349,18351,18354],{"className":18350},[61],[17,18352,4935],{"className":18353},[61,62],[17,18355,18357],{"className":18356},[433],[17,18358,18360,18380],{"className":18359},[437,1313],[17,18361,18363,18377],{"className":18362},[441],[17,18364,18366],{"className":18365,"style":18165},[445],[17,18367,18368,18371],{"style":1407},[17,18369],{"className":18370,"style":453},[452],[17,18372,18374],{"className":18373},[457,458,459,460],[17,18375,18025],{"className":18376},[61,62,460],[17,18378,1346],{"className":18379},[1345],[17,18381,18383],{"className":18382},[441],[17,18384,18386],{"className":18385,"style":1429},[445],[17,18387],{},". The theorem claims that:",[17,18390,18392],{"className":18391,"translate":21},[807],[17,18393,18395,18455],{"className":18394,"translate":21},[20],[17,18396,18398],{"className":18397},[25],[27,18399,18400],{"xmlns":29,"display":816},[31,18401,18402,18452],{},[34,18403,18404,18406,18408,18414,18420,18422,18428,18434,18436,18438,18440,18446],{},[37,18405,111],{},[204,18407,825],{},[848,18409,18410,18412],{},[37,18411,172],{},[408,18413,3544],{},[848,18415,18416,18418],{},[37,18417,4935],{},[408,18419,3544],{},[204,18421,3557],{},[848,18423,18424,18426],{},[37,18425,172],{},[408,18427,410],{},[848,18429,18430,18432],{},[37,18431,4935],{},[408,18433,410],{},[204,18435,3557],{},[204,18437,3905],{},[204,18439,3557],{},[848,18441,18442,18444],{},[37,18443,172],{},[37,18445,18025],{},[848,18447,18448,18450],{},[37,18449,4935],{},[37,18451,18025],{},[41,18453,18454],{"encoding":43},"D = M_1A_1 + M_2A_2 + \\cdots + M_tA_t",[17,18456,18458,18476,18571,18666,18684],{"className":18457,"ariaHidden":48},[47],[17,18459,18461,18464,18467,18470,18473],{"className":18460},[52],[17,18462],{"className":18463,"style":123},[56],[17,18465,111],{"className":18466,"style":127},[61,62],[17,18468],{"className":18469,"style":372},[228],[17,18471,825],{"className":18472},[376],[17,18474],{"className":18475,"style":372},[228],[17,18477,18479,18482,18522,18562,18565,18568],{"className":18478},[52],[17,18480],{"className":18481,"style":14691},[56],[17,18483,18485,18488],{"className":18484},[61],[17,18486,172],{"className":18487,"style":187},[61,62],[17,18489,18491],{"className":18490},[433],[17,18492,18494,18514],{"className":18493},[437,1313],[17,18495,18497,18511],{"className":18496},[441],[17,18498,18500],{"className":18499,"style":1404},[445],[17,18501,18502,18505],{"style":18059},[17,18503],{"className":18504,"style":453},[452],[17,18506,18508],{"className":18507},[457,458,459,460],[17,18509,3544],{"className":18510},[61,460],[17,18512,1346],{"className":18513},[1345],[17,18515,18517],{"className":18516},[441],[17,18518,18520],{"className":18519,"style":1429},[445],[17,18521],{},[17,18523,18525,18528],{"className":18524},[61],[17,18526,4935],{"className":18527},[61,62],[17,18529,18531],{"className":18530},[433],[17,18532,18534,18554],{"className":18533},[437,1313],[17,18535,18537,18551],{"className":18536},[441],[17,18538,18540],{"className":18539,"style":1404},[445],[17,18541,18542,18545],{"style":1407},[17,18543],{"className":18544,"style":453},[452],[17,18546,18548],{"className":18547},[457,458,459,460],[17,18549,3544],{"className":18550},[61,460],[17,18552,1346],{"className":18553},[1345],[17,18555,18557],{"className":18556},[441],[17,18558,18560],{"className":18559,"style":1429},[445],[17,18561],{},[17,18563],{"className":18564,"style":229},[228],[17,18566,3557],{"className":18567},[233],[17,18569],{"className":18570,"style":229},[228],[17,18572,18574,18577,18617,18657,18660,18663],{"className":18573},[52],[17,18575],{"className":18576,"style":14691},[56],[17,18578,18580,18583],{"className":18579},[61],[17,18581,172],{"className":18582,"style":187},[61,62],[17,18584,18586],{"className":18585},[433],[17,18587,18589,18609],{"className":18588},[437,1313],[17,18590,18592,18606],{"className":18591},[441],[17,18593,18595],{"className":18594,"style":1404},[445],[17,18596,18597,18600],{"style":18059},[17,18598],{"className":18599,"style":453},[452],[17,18601,18603],{"className":18602},[457,458,459,460],[17,18604,410],{"className":18605},[61,460],[17,18607,1346],{"className":18608},[1345],[17,18610,18612],{"className":18611},[441],[17,18613,18615],{"className":18614,"style":1429},[445],[17,18616],{},[17,18618,18620,18623],{"className":18619},[61],[17,18621,4935],{"className":18622},[61,62],[17,18624,18626],{"className":18625},[433],[17,18627,18629,18649],{"className":18628},[437,1313],[17,18630,18632,18646],{"className":18631},[441],[17,18633,18635],{"className":18634,"style":1404},[445],[17,18636,18637,18640],{"style":1407},[17,18638],{"className":18639,"style":453},[452],[17,18641,18643],{"className":18642},[457,458,459,460],[17,18644,410],{"className":18645},[61,460],[17,18647,1346],{"className":18648},[1345],[17,18650,18652],{"className":18651},[441],[17,18653,18655],{"className":18654,"style":1429},[445],[17,18656],{},[17,18658],{"className":18659,"style":229},[228],[17,18661,3557],{"className":18662},[233],[17,18664],{"className":18665,"style":229},[228],[17,18667,18669,18672,18675,18678,18681],{"className":18668},[52],[17,18670],{"className":18671,"style":13282},[56],[17,18673,3905],{"className":18674},[1301],[17,18676],{"className":18677,"style":229},[228],[17,18679,3557],{"className":18680},[233],[17,18682],{"className":18683,"style":229},[228],[17,18685,18687,18690,18730],{"className":18686},[52],[17,18688],{"className":18689,"style":14691},[56],[17,18691,18693,18696],{"className":18692},[61],[17,18694,172],{"className":18695,"style":187},[61,62],[17,18697,18699],{"className":18698},[433],[17,18700,18702,18722],{"className":18701},[437,1313],[17,18703,18705,18719],{"className":18704},[441],[17,18706,18708],{"className":18707,"style":18165},[445],[17,18709,18710,18713],{"style":18059},[17,18711],{"className":18712,"style":453},[452],[17,18714,18716],{"className":18715},[457,458,459,460],[17,18717,18025],{"className":18718},[61,62,460],[17,18720,1346],{"className":18721},[1345],[17,18723,18725],{"className":18724},[441],[17,18726,18728],{"className":18727,"style":1429},[445],[17,18729],{},[17,18731,18733,18736],{"className":18732},[61],[17,18734,4935],{"className":18735},[61,62],[17,18737,18739],{"className":18738},[433],[17,18740,18742,18762],{"className":18741},[437,1313],[17,18743,18745,18759],{"className":18744},[441],[17,18746,18748],{"className":18747,"style":18165},[445],[17,18749,18750,18753],{"style":1407},[17,18751],{"className":18752,"style":453},[452],[17,18754,18756],{"className":18755},[457,458,459,460],[17,18757,18025],{"className":18758},[61,62,460],[17,18760,1346],{"className":18761},[1345],[17,18763,18765],{"className":18764},[441],[17,18766,18768],{"className":18767,"style":1429},[445],[17,18769],{},[10,18771,18772],{},"where",[17,18774,18776],{"className":18775,"translate":21},[807],[17,18777,18779,18837],{"className":18778,"translate":21},[20],[17,18780,18782],{"className":18781},[25],[27,18783,18784],{"xmlns":29,"display":816},[31,18785,18786,18834],{},[34,18787,18788,18790,18792,18802,18804],{},[37,18789,18025],{},[204,18791,825],{},[18793,18794,18795,18798,18800],"msubsup",{},[3907,18796,18797],{},"C",[37,18799,80],{},[37,18801,39],{},[204,18803,825],{},[844,18805,18806],{"scriptlevel":846,"displaystyle":48},[18807,18808,18809,18816],"mfrac",{},[34,18810,18811,18813],{},[37,18812,80],{},[204,18814,18815],{"stretchy":621},"!",[34,18817,18818,18820,18822,18824,18826,18828,18830,18832],{},[37,18819,39],{},[204,18821,18815],{"stretchy":621},[204,18823,622],{"stretchy":621},[37,18825,80],{},[204,18827,627],{},[37,18829,39],{},[204,18831,632],{"stretchy":621},[204,18833,18815],{"stretchy":621},[41,18835,18836],{"encoding":43},"t = \\text{C}_{n}^{k} = \\dfrac{n!}{k!(n-k)!}",[17,18838,18840,18859,18940],{"className":18839,"ariaHidden":48},[47],[17,18841,18843,18847,18850,18853,18856],{"className":18842},[52],[17,18844],{"className":18845,"style":18846},[56],"height:0.6151em;",[17,18848,18025],{"className":18849},[61,62],[17,18851],{"className":18852,"style":372},[228],[17,18854,825],{"className":18855},[376],[17,18857],{"className":18858,"style":372},[228],[17,18860,18862,18866,18931,18934,18937],{"className":18861},[52],[17,18863],{"className":18864,"style":18865},[56],"height:1.1693em;vertical-align:-0.247em;",[17,18867,18869,18876],{"className":18868},[61],[17,18870,18873],{"className":18871},[61,18872],"text",[17,18874,18797],{"className":18875},[61],[17,18877,18879],{"className":18878},[433],[17,18880,18882,18922],{"className":18881},[437,1313],[17,18883,18885,18919],{"className":18884},[441],[17,18886,18889,18904],{"className":18887,"style":18888},[445],"height:0.9223em;",[17,18890,18892,18895],{"style":18891},"top:-2.453em;margin-right:0.05em;",[17,18893],{"className":18894,"style":453},[452],[17,18896,18898],{"className":18897},[457,458,459,460],[17,18899,18901],{"className":18900},[61,460],[17,18902,80],{"className":18903},[61,62,460],[17,18905,18907,18910],{"style":18906},"top:-3.1362em;margin-right:0.05em;",[17,18908],{"className":18909,"style":453},[452],[17,18911,18913],{"className":18912},[457,458,459,460],[17,18914,18916],{"className":18915},[61,460],[17,18917,39],{"className":18918,"style":63},[61,62,460],[17,18920,1346],{"className":18921},[1345],[17,18923,18925],{"className":18924},[441],[17,18926,18929],{"className":18927,"style":18928},[445],"height:0.247em;",[17,18930],{},[17,18932],{"className":18933,"style":372},[228],[17,18935,825],{"className":18936},[376],[17,18938],{"className":18939,"style":372},[228],[17,18941,18943,18947],{"className":18942},[52],[17,18944],{"className":18945,"style":18946},[56],"height:2.3074em;vertical-align:-0.936em;",[17,18948,18950,18954,19042],{"className":18949},[61],[17,18951],{"className":18952},[649,18953],"nulldelimiter",[17,18955,18957],{"className":18956},[18807],[17,18958,18960,19033],{"className":18959},[437,1313],[17,18961,18963,19030],{"className":18962},[441],[17,18964,18967,19004,19015],{"className":18965,"style":18966},[445],"height:1.3714em;",[17,18968,18970,18973],{"style":18969},"top:-2.314em;",[17,18971],{"className":18972,"style":2511},[452],[17,18974,18976,18979,18982,18985,18988,18991,18994,18997,19000],{"className":18975},[61],[17,18977,39],{"className":18978,"style":63},[61,62],[17,18980,18815],{"className":18981},[674],[17,18983,622],{"className":18984},[649],[17,18986,80],{"className":18987},[61,62],[17,18989],{"className":18990,"style":229},[228],[17,18992,627],{"className":18993},[233],[17,18995],{"className":18996,"style":229},[228],[17,18998,39],{"className":18999,"style":63},[61,62],[17,19001,19003],{"className":19002},[674],")!",[17,19005,19007,19010],{"style":19006},"top:-3.23em;",[17,19008],{"className":19009,"style":2511},[452],[17,19011],{"className":19012,"style":19014},[19013],"frac-line","border-bottom-width:0.04em;",[17,19016,19018,19021],{"style":19017},"top:-3.677em;",[17,19019],{"className":19020,"style":2511},[452],[17,19022,19024,19027],{"className":19023},[61],[17,19025,80],{"className":19026},[61,62],[17,19028,18815],{"className":19029},[674],[17,19031,1346],{"className":19032},[1345],[17,19034,19036],{"className":19035},[441],[17,19037,19040],{"className":19038,"style":19039},[445],"height:0.936em;",[17,19041],{},[17,19043],{"className":19044},[674,18953],[10,19046,19047,19048,16553,19165,19193,19194,19309,19310,333,19427,19519],{},"By the lemma, each term in ",[17,19049,19051,19075],{"className":19050,"translate":21},[20],[17,19052,19054],{"className":19053},[25],[27,19055,19056],{"xmlns":29},[31,19057,19058,19072],{},[34,19059,19060,19066],{},[848,19061,19062,19064],{},[37,19063,172],{},[37,19065,3890],{},[848,19067,19068,19070],{},[37,19069,4935],{},[37,19071,3890],{},[41,19073,19074],{"encoding":43},"M_iA_i",[17,19076,19078],{"className":19077,"ariaHidden":48},[47],[17,19079,19081,19084,19125],{"className":19080},[52],[17,19082],{"className":19083,"style":14691},[56],[17,19085,19087,19090],{"className":19086},[61],[17,19088,172],{"className":19089,"style":187},[61,62],[17,19091,19093],{"className":19092},[433],[17,19094,19096,19117],{"className":19095},[437,1313],[17,19097,19099,19114],{"className":19098},[441],[17,19100,19103],{"className":19101,"style":19102},[445],"height:0.3117em;",[17,19104,19105,19108],{"style":18059},[17,19106],{"className":19107,"style":453},[452],[17,19109,19111],{"className":19110},[457,458,459,460],[17,19112,3890],{"className":19113},[61,62,460],[17,19115,1346],{"className":19116},[1345],[17,19118,19120],{"className":19119},[441],[17,19121,19123],{"className":19122,"style":1429},[445],[17,19124],{},[17,19126,19128,19131],{"className":19127},[61],[17,19129,4935],{"className":19130},[61,62],[17,19132,19134],{"className":19133},[433],[17,19135,19137,19157],{"className":19136},[437,1313],[17,19138,19140,19154],{"className":19139},[441],[17,19141,19143],{"className":19142,"style":19102},[445],[17,19144,19145,19148],{"style":1407},[17,19146],{"className":19147,"style":453},[452],[17,19149,19151],{"className":19150},[457,458,459,460],[17,19152,3890],{"className":19153},[61,62,460],[17,19155,1346],{"className":19156},[1345],[17,19158,19160],{"className":19159},[441],[17,19161,19163],{"className":19162,"style":1429},[445],[17,19164],{},[17,19166,19168,19181],{"className":19167,"translate":21},[20],[17,19169,19171],{"className":19170},[25],[27,19172,19173],{"xmlns":29},[31,19174,19175,19179],{},[34,19176,19177],{},[37,19178,111],{},[41,19180,111],{"encoding":43},[17,19182,19184],{"className":19183,"ariaHidden":48},[47],[17,19185,19187,19190],{"className":19186},[52],[17,19188],{"className":19189,"style":123},[56],[17,19191,111],{"className":19192,"style":127},[61,62]," with the same sign, and there are no common terms between ",[17,19195,19197,19220],{"className":19196,"translate":21},[20],[17,19198,19200],{"className":19199},[25],[27,19201,19202],{"xmlns":29},[31,19203,19204,19218],{},[34,19205,19206,19212],{},[848,19207,19208,19210],{},[37,19209,172],{},[37,19211,3890],{},[848,19213,19214,19216],{},[37,19215,4935],{},[37,19217,3890],{},[41,19219,19074],{"encoding":43},[17,19221,19223],{"className":19222,"ariaHidden":48},[47],[17,19224,19226,19229,19269],{"className":19225},[52],[17,19227],{"className":19228,"style":14691},[56],[17,19230,19232,19235],{"className":19231},[61],[17,19233,172],{"className":19234,"style":187},[61,62],[17,19236,19238],{"className":19237},[433],[17,19239,19241,19261],{"className":19240},[437,1313],[17,19242,19244,19258],{"className":19243},[441],[17,19245,19247],{"className":19246,"style":19102},[445],[17,19248,19249,19252],{"style":18059},[17,19250],{"className":19251,"style":453},[452],[17,19253,19255],{"className":19254},[457,458,459,460],[17,19256,3890],{"className":19257},[61,62,460],[17,19259,1346],{"className":19260},[1345],[17,19262,19264],{"className":19263},[441],[17,19265,19267],{"className":19266,"style":1429},[445],[17,19268],{},[17,19270,19272,19275],{"className":19271},[61],[17,19273,4935],{"className":19274},[61,62],[17,19276,19278],{"className":19277},[433],[17,19279,19281,19301],{"className":19280},[437,1313],[17,19282,19284,19298],{"className":19283},[441],[17,19285,19287],{"className":19286,"style":19102},[445],[17,19288,19289,19292],{"style":1407},[17,19290],{"className":19291,"style":453},[452],[17,19293,19295],{"className":19294},[457,458,459,460],[17,19296,3890],{"className":19297},[61,62,460],[17,19299,1346],{"className":19300},[1345],[17,19302,19304],{"className":19303},[441],[17,19305,19307],{"className":19306,"style":1429},[445],[17,19308],{}," and ",[17,19311,19313,19337],{"className":19312,"translate":21},[20],[17,19314,19316],{"className":19315},[25],[27,19317,19318],{"xmlns":29},[31,19319,19320,19334],{},[34,19321,19322,19328],{},[848,19323,19324,19326],{},[37,19325,172],{},[37,19327,3928],{},[848,19329,19330,19332],{},[37,19331,4935],{},[37,19333,3928],{},[41,19335,19336],{"encoding":43},"M_jA_j",[17,19338,19340],{"className":19339,"ariaHidden":48},[47],[17,19341,19343,19347,19387],{"className":19342},[52],[17,19344],{"className":19345,"style":19346},[56],"height:0.9694em;vertical-align:-0.2861em;",[17,19348,19350,19353],{"className":19349},[61],[17,19351,172],{"className":19352,"style":187},[61,62],[17,19354,19356],{"className":19355},[433],[17,19357,19359,19379],{"className":19358},[437,1313],[17,19360,19362,19376],{"className":19361},[441],[17,19363,19365],{"className":19364,"style":19102},[445],[17,19366,19367,19370],{"style":18059},[17,19368],{"className":19369,"style":453},[452],[17,19371,19373],{"className":19372},[457,458,459,460],[17,19374,3928],{"className":19375,"style":4138},[61,62,460],[17,19377,1346],{"className":19378},[1345],[17,19380,19382],{"className":19381},[441],[17,19383,19385],{"className":19384,"style":5963},[445],[17,19386],{},[17,19388,19390,19393],{"className":19389},[61],[17,19391,4935],{"className":19392},[61,62],[17,19394,19396],{"className":19395},[433],[17,19397,19399,19419],{"className":19398},[437,1313],[17,19400,19402,19416],{"className":19401},[441],[17,19403,19405],{"className":19404,"style":19102},[445],[17,19406,19407,19410],{"style":1407},[17,19408],{"className":19409,"style":453},[452],[17,19411,19413],{"className":19412},[457,458,459,460],[17,19414,3928],{"className":19415,"style":4138},[61,62,460],[17,19417,1346],{"className":19418},[1345],[17,19420,19422],{"className":19421},[441],[17,19423,19425],{"className":19424,"style":5963},[445],[17,19426],{},[17,19428,19430,19449],{"className":19429,"translate":21},[20],[17,19431,19433],{"className":19432},[25],[27,19434,19435],{"xmlns":29},[31,19436,19437,19446],{},[34,19438,19439,19441,19444],{},[37,19440,3890],{},[204,19442,19443],{"mathvariant":693},"≠",[37,19445,3928],{},[41,19447,19448],{"encoding":43},"i \\ne j",[17,19450,19452,19510],{"className":19451,"ariaHidden":48},[47],[17,19453,19455,19458,19461,19464,19507],{"className":19454},[52],[17,19456],{"className":19457,"style":8579},[56],[17,19459,3890],{"className":19460},[61,62],[17,19462],{"className":19463,"style":372},[228],[17,19465,19467,19500,19504],{"className":19466},[376],[17,19468,19470],{"className":19469},[376],[17,19471,19474],{"className":19472},[61,19473],"vbox",[17,19475,19478],{"className":19476},[19477],"thinbox",[17,19479,19482,19485,19496],{"className":19480},[19481],"rlap",[17,19483],{"className":19484,"style":8579},[56],[17,19486,19489],{"className":19487},[19488],"inner",[17,19490,19492],{"className":19491},[61],[17,19493,19495],{"className":19494},[376],"",[17,19497],{"className":19498},[19499],"fix",[17,19501],{"className":19502},[228,19503],"nobreak",[17,19505,825],{"className":19506},[376],[17,19508],{"className":19509,"style":372},[228],[17,19511,19513,19516],{"className":19512},[52],[17,19514],{"className":19515,"style":3965},[56],[17,19517,3928],{"className":19518,"style":4138},[61,62],").",[10,19521,19522,19523,19557],{},"The left-hand side of the equation contains ",[17,19524,19526,19542],{"className":19525,"translate":21},[20],[17,19527,19529],{"className":19528},[25],[27,19530,19531],{"xmlns":29},[31,19532,19533,19539],{},[34,19534,19535,19537],{},[37,19536,80],{},[204,19538,18815],{"stretchy":621},[41,19540,19541],{"encoding":43},"n!",[17,19543,19545],{"className":19544,"ariaHidden":48},[47],[17,19546,19548,19551,19554],{"className":19547},[52],[17,19549],{"className":19550,"style":57},[56],[17,19552,80],{"className":19553},[61,62],[17,19555,18815],{"className":19556},[674]," terms, and the right-hand side contains",[10,19559,19560],{},[17,19561,19563,19658],{"className":19562,"translate":21},[20],[17,19564,19566],{"className":19565},[25],[27,19567,19568],{"xmlns":29},[31,19569,19570,19655],{},[34,19571,19572,19587,19589,19591,19593,19595,19597,19599,19601,19603,19605,19607,19609,19611,19613,19615,19617,19619,19621,19649,19651,19653],{},[18793,19573,19574,19577,19585],{},[204,19575,19576],{},"∑",[34,19578,19579,19581,19583],{},[37,19580,3890],{},[204,19582,825],{},[408,19584,3544],{},[37,19586,18025],{},[37,19588,39],{},[204,19590,18815],{"stretchy":621},[204,19592,622],{"stretchy":621},[37,19594,80],{},[204,19596,627],{},[37,19598,39],{},[204,19600,632],{"stretchy":621},[204,19602,18815],{"stretchy":621},[204,19604,825],{},[37,19606,39],{},[204,19608,18815],{"stretchy":621},[204,19610,622],{"stretchy":621},[37,19612,80],{},[204,19614,627],{},[37,19616,39],{},[204,19618,632],{"stretchy":621},[204,19620,18815],{"stretchy":621},[844,19622,19623],{"scriptlevel":846,"displaystyle":48},[18807,19624,19625,19631],{},[34,19626,19627,19629],{},[37,19628,80],{},[204,19630,18815],{"stretchy":621},[34,19632,19633,19635,19637,19639,19641,19643,19645,19647],{},[37,19634,39],{},[204,19636,18815],{"stretchy":621},[204,19638,622],{"stretchy":621},[37,19640,80],{},[204,19642,627],{},[37,19644,39],{},[204,19646,632],{"stretchy":621},[204,19648,18815],{"stretchy":621},[204,19650,825],{},[37,19652,80],{},[204,19654,18815],{"stretchy":621},[41,19656,19657],{"encoding":43},"\\sum\\limits_{i=1}^{t}k!(n-k)! = k!(n-k)!\\dfrac{n!}{k!(n-k)!} = n!",[17,19659,19661,19768,19789,19816,19926],{"className":19660,"ariaHidden":48},[47],[17,19662,19664,19668,19744,19747,19750,19753,19756,19759,19762,19765],{"className":19663},[52],[17,19665],{"className":19666,"style":19667},[56],"height:2.4582em;vertical-align:-0.9777em;",[17,19669,19673],{"className":19670},[19671,19672],"mop","op-limits",[17,19674,19676,19735],{"className":19675},[437,1313],[17,19677,19679,19732],{"className":19678},[441],[17,19680,19683,19704,19717],{"className":19681,"style":19682},[445],"height:1.4806em;",[17,19684,19686,19689],{"style":19685},"top:-2.1223em;margin-left:0em;",[17,19687],{"className":19688,"style":2511},[452],[17,19690,19692],{"className":19691},[457,458,459,460],[17,19693,19695,19698,19701],{"className":19694},[61,460],[17,19696,3890],{"className":19697},[61,62,460],[17,19699,825],{"className":19700},[376,460],[17,19702,3544],{"className":19703},[61,460],[17,19705,19707,19710],{"style":19706},"top:-3em;",[17,19708],{"className":19709,"style":2511},[452],[17,19711,19712],{},[17,19713,19576],{"className":19714},[19671,19715,19716],"op-symbol","small-op",[17,19718,19720,19723],{"style":19719},"top:-3.95em;margin-left:0em;",[17,19721],{"className":19722,"style":2511},[452],[17,19724,19726],{"className":19725},[457,458,459,460],[17,19727,19729],{"className":19728},[61,460],[17,19730,18025],{"className":19731},[61,62,460],[17,19733,1346],{"className":19734},[1345],[17,19736,19738],{"className":19737},[441],[17,19739,19742],{"className":19740,"style":19741},[445],"height:0.9777em;",[17,19743],{},[17,19745],{"className":19746,"style":2406},[228],[17,19748,39],{"className":19749,"style":63},[61,62],[17,19751,18815],{"className":19752},[674],[17,19754,622],{"className":19755},[649],[17,19757,80],{"className":19758},[61,62],[17,19760],{"className":19761,"style":229},[228],[17,19763,627],{"className":19764},[233],[17,19766],{"className":19767,"style":229},[228],[17,19769,19771,19774,19777,19780,19783,19786],{"className":19770},[52],[17,19772],{"className":19773,"style":645},[56],[17,19775,39],{"className":19776,"style":63},[61,62],[17,19778,19003],{"className":19779},[674],[17,19781],{"className":19782,"style":372},[228],[17,19784,825],{"className":19785},[376],[17,19787],{"className":19788,"style":372},[228],[17,19790,19792,19795,19798,19801,19804,19807,19810,19813],{"className":19791},[52],[17,19793],{"className":19794,"style":645},[56],[17,19796,39],{"className":19797,"style":63},[61,62],[17,19799,18815],{"className":19800},[674],[17,19802,622],{"className":19803},[649],[17,19805,80],{"className":19806},[61,62],[17,19808],{"className":19809,"style":229},[228],[17,19811,627],{"className":19812},[233],[17,19814],{"className":19815,"style":229},[228],[17,19817,19819,19822,19825,19828,19917,19920,19923],{"className":19818},[52],[17,19820],{"className":19821,"style":18946},[56],[17,19823,39],{"className":19824,"style":63},[61,62],[17,19826,19003],{"className":19827},[674],[17,19829,19831,19834,19914],{"className":19830},[61],[17,19832],{"className":19833},[649,18953],[17,19835,19837],{"className":19836},[18807],[17,19838,19840,19906],{"className":19839},[437,1313],[17,19841,19843,19903],{"className":19842},[441],[17,19844,19846,19881,19889],{"className":19845,"style":18966},[445],[17,19847,19848,19851],{"style":18969},[17,19849],{"className":19850,"style":2511},[452],[17,19852,19854,19857,19860,19863,19866,19869,19872,19875,19878],{"className":19853},[61],[17,19855,39],{"className":19856,"style":63},[61,62],[17,19858,18815],{"className":19859},[674],[17,19861,622],{"className":19862},[649],[17,19864,80],{"className":19865},[61,62],[17,19867],{"className":19868,"style":229},[228],[17,19870,627],{"className":19871},[233],[17,19873],{"className":19874,"style":229},[228],[17,19876,39],{"className":19877,"style":63},[61,62],[17,19879,19003],{"className":19880},[674],[17,19882,19883,19886],{"style":19006},[17,19884],{"className":19885,"style":2511},[452],[17,19887],{"className":19888,"style":19014},[19013],[17,19890,19891,19894],{"style":19017},[17,19892],{"className":19893,"style":2511},[452],[17,19895,19897,19900],{"className":19896},[61],[17,19898,80],{"className":19899},[61,62],[17,19901,18815],{"className":19902},[674],[17,19904,1346],{"className":19905},[1345],[17,19907,19909],{"className":19908},[441],[17,19910,19912],{"className":19911,"style":19039},[445],[17,19913],{},[17,19915],{"className":19916},[674,18953],[17,19918],{"className":19919,"style":372},[228],[17,19921,825],{"className":19922},[376],[17,19924],{"className":19925,"style":372},[228],[17,19927,19929,19932,19935],{"className":19928},[52],[17,19930],{"className":19931,"style":57},[56],[17,19933,80],{"className":19934},[61,62],[17,19936,18815],{"className":19937},[674],[10,19939,19940],{},"terms. Hence, the theorem is proven.",[10,19942,19943,19945,19946,20492],{},[13,19944,803],{}," In the determinant ",[17,19947,19949,20082],{"className":19948,"translate":21},[20],[17,19950,19952],{"className":19951},[25],[27,19953,19954],{"xmlns":29},[31,19955,19956,20079],{},[34,19957,19958,19960,19962],{},[37,19959,111],{},[204,19961,825],{},[34,19963,19964,19966,20077],{},[204,19965,830],{"fence":48},[832,19967,19969,19995,20025,20051],{"rowspacing":834,"columnalign":19968,"columnspacing":836},"center center center center",[838,19970,19971,19977,19983,19989],{},[841,19972,19973],{},[844,19974,19975],{"scriptlevel":846,"displaystyle":621},[408,19976,3544],{},[841,19978,19979],{},[844,19980,19981],{"scriptlevel":846,"displaystyle":621},[408,19982,410],{},[841,19984,19985],{},[844,19986,19987],{"scriptlevel":846,"displaystyle":621},[408,19988,3544],{},[841,19990,19991],{},[844,19992,19993],{"scriptlevel":846,"displaystyle":621},[408,19994,3564],{},[838,19996,19997,20003,20013,20019],{},[841,19998,19999],{},[844,20000,20001],{"scriptlevel":846,"displaystyle":621},[408,20002,846],{},[841,20004,20005],{},[844,20006,20007],{"scriptlevel":846,"displaystyle":621},[34,20008,20009,20011],{},[204,20010,627],{},[408,20012,3544],{},[841,20014,20015],{},[844,20016,20017],{"scriptlevel":846,"displaystyle":621},[408,20018,410],{},[841,20020,20021],{},[844,20022,20023],{"scriptlevel":846,"displaystyle":621},[408,20024,3544],{},[838,20026,20027,20033,20039,20045],{},[841,20028,20029],{},[844,20030,20031],{"scriptlevel":846,"displaystyle":621},[408,20032,3544],{},[841,20034,20035],{},[844,20036,20037],{"scriptlevel":846,"displaystyle":621},[408,20038,846],{},[841,20040,20041],{},[844,20042,20043],{"scriptlevel":846,"displaystyle":621},[408,20044,3544],{},[841,20046,20047],{},[844,20048,20049],{"scriptlevel":846,"displaystyle":621},[408,20050,3577],{},[838,20052,20053,20059,20065,20071],{},[841,20054,20055],{},[844,20056,20057],{"scriptlevel":846,"displaystyle":621},[408,20058,846],{},[841,20060,20061],{},[844,20062,20063],{"scriptlevel":846,"displaystyle":621},[408,20064,3544],{},[841,20066,20067],{},[844,20068,20069],{"scriptlevel":846,"displaystyle":621},[408,20070,3577],{},[841,20072,20073],{},[844,20074,20075],{"scriptlevel":846,"displaystyle":621},[408,20076,3544],{},[204,20078,830],{"fence":48},[41,20080,20081],{"encoding":43},"D=\\begin{vmatrix} 1  & 2 & 1 &4 \\\\  0 & -1 & 2 & 1\\\\  1 & 0 & 1 & 3\\\\ 0  &  1& 3 & 1\\end{vmatrix}",[17,20083,20085,20103],{"className":20084,"ariaHidden":48},[47],[17,20086,20088,20091,20094,20097,20100],{"className":20087},[52],[17,20089],{"className":20090,"style":123},[56],[17,20092,111],{"className":20093,"style":127},[61,62],[17,20095],{"className":20096,"style":372},[228],[17,20098,825],{"className":20099},[376],[17,20101],{"className":20102,"style":372},[228],[17,20104,20106,20110],{"className":20105},[52],[17,20107],{"className":20108,"style":20109},[56],"height:4.8em;vertical-align:-2.15em;",[17,20111,20113,20158,20455],{"className":20112},[1301],[17,20114,20116],{"className":20115},[649],[17,20117,20119],{"className":20118},[1308,1309],[17,20120,20122,20149],{"className":20121},[437,1313],[17,20123,20125,20146],{"className":20124},[441],[17,20126,20129],{"className":20127,"style":20128},[445],"height:2.65em;",[17,20130,20132,20136],{"style":20131},"top:-4.65em;",[17,20133],{"className":20134,"style":20135},[452],"height:6.8em;",[17,20137,20139],{"style":20138},"width:0.333em;height:4.800em;",[1332,20140,20143],{"xmlns":1334,"width":1335,"height":20141,"viewBox":20142},"4.800em","0 0 333 4800",[1339,20144],{"d":20145},"M145 15 v585 v3600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-3600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v3600 v585 h43z",[17,20147,1346],{"className":20148},[1345],[17,20150,20152],{"className":20151},[441],[17,20153,20156],{"className":20154,"style":20155},[445],"height:2.15em;",[17,20157],{},[17,20159,20161],{"className":20160},[61],[17,20162,20164,20233,20236,20239,20309,20312,20315,20382,20385,20388],{"className":20163},[832],[17,20165,20167],{"className":20166},[1365],[17,20168,20170,20225],{"className":20169},[437,1313],[17,20171,20173,20222],{"className":20172},[441],[17,20174,20176,20188,20199,20210],{"className":20175,"style":20128},[445],[17,20177,20179,20182],{"style":20178},"top:-4.81em;",[17,20180],{"className":20181,"style":2511},[452],[17,20183,20185],{"className":20184},[61],[17,20186,3544],{"className":20187},[61],[17,20189,20190,20193],{"style":3207},[17,20191],{"className":20192,"style":2511},[452],[17,20194,20196],{"className":20195},[61],[17,20197,846],{"className":20198},[61],[17,20200,20201,20204],{"style":3259},[17,20202],{"className":20203,"style":2511},[452],[17,20205,20207],{"className":20206},[61],[17,20208,3544],{"className":20209},[61],[17,20211,20213,20216],{"style":20212},"top:-1.21em;",[17,20214],{"className":20215,"style":2511},[452],[17,20217,20219],{"className":20218},[61],[17,20220,846],{"className":20221},[61],[17,20223,1346],{"className":20224},[1345],[17,20226,20228],{"className":20227},[441],[17,20229,20231],{"className":20230,"style":20155},[445],[17,20232],{},[17,20234],{"className":20235,"style":1572},[1571],[17,20237],{"className":20238,"style":1572},[1571],[17,20240,20242],{"className":20241},[1365],[17,20243,20245,20301],{"className":20244},[437,1313],[17,20246,20248,20298],{"className":20247},[441],[17,20249,20251,20262,20276,20287],{"className":20250,"style":20128},[445],[17,20252,20253,20256],{"style":20178},[17,20254],{"className":20255,"style":2511},[452],[17,20257,20259],{"className":20258},[61],[17,20260,410],{"className":20261},[61],[17,20263,20264,20267],{"style":3207},[17,20265],{"className":20266,"style":2511},[452],[17,20268,20270,20273],{"className":20269},[61],[17,20271,627],{"className":20272},[61],[17,20274,3544],{"className":20275},[61],[17,20277,20278,20281],{"style":3259},[17,20279],{"className":20280,"style":2511},[452],[17,20282,20284],{"className":20283},[61],[17,20285,846],{"className":20286},[61],[17,20288,20289,20292],{"style":20212},[17,20290],{"className":20291,"style":2511},[452],[17,20293,20295],{"className":20294},[61],[17,20296,3544],{"className":20297},[61],[17,20299,1346],{"className":20300},[1345],[17,20302,20304],{"className":20303},[441],[17,20305,20307],{"className":20306,"style":20155},[445],[17,20308],{},[17,20310],{"className":20311,"style":1572},[1571],[17,20313],{"className":20314,"style":1572},[1571],[17,20316,20318],{"className":20317},[1365],[17,20319,20321,20374],{"className":20320},[437,1313],[17,20322,20324,20371],{"className":20323},[441],[17,20325,20327,20338,20349,20360],{"className":20326,"style":20128},[445],[17,20328,20329,20332],{"style":20178},[17,20330],{"className":20331,"style":2511},[452],[17,20333,20335],{"className":20334},[61],[17,20336,3544],{"className":20337},[61],[17,20339,20340,20343],{"style":3207},[17,20341],{"className":20342,"style":2511},[452],[17,20344,20346],{"className":20345},[61],[17,20347,410],{"className":20348},[61],[17,20350,20351,20354],{"style":3259},[17,20352],{"className":20353,"style":2511},[452],[17,20355,20357],{"className":20356},[61],[17,20358,3544],{"className":20359},[61],[17,20361,20362,20365],{"style":20212},[17,20363],{"className":20364,"style":2511},[452],[17,20366,20368],{"className":20367},[61],[17,20369,3577],{"className":20370},[61],[17,20372,1346],{"className":20373},[1345],[17,20375,20377],{"className":20376},[441],[17,20378,20380],{"className":20379,"style":20155},[445],[17,20381],{},[17,20383],{"className":20384,"style":1572},[1571],[17,20386],{"className":20387,"style":1572},[1571],[17,20389,20391],{"className":20390},[1365],[17,20392,20394,20447],{"className":20393},[437,1313],[17,20395,20397,20444],{"className":20396},[441],[17,20398,20400,20411,20422,20433],{"className":20399,"style":20128},[445],[17,20401,20402,20405],{"style":20178},[17,20403],{"className":20404,"style":2511},[452],[17,20406,20408],{"className":20407},[61],[17,20409,3564],{"className":20410},[61],[17,20412,20413,20416],{"style":3207},[17,20414],{"className":20415,"style":2511},[452],[17,20417,20419],{"className":20418},[61],[17,20420,3544],{"className":20421},[61],[17,20423,20424,20427],{"style":3259},[17,20425],{"className":20426,"style":2511},[452],[17,20428,20430],{"className":20429},[61],[17,20431,3577],{"className":20432},[61],[17,20434,20435,20438],{"style":20212},[17,20436],{"className":20437,"style":2511},[452],[17,20439,20441],{"className":20440},[61],[17,20442,3544],{"className":20443},[61],[17,20445,1346],{"className":20446},[1345],[17,20448,20450],{"className":20449},[441],[17,20451,20453],{"className":20452,"style":20155},[445],[17,20454],{},[17,20456,20458],{"className":20457},[674],[17,20459,20461],{"className":20460},[1308,1309],[17,20462,20464,20484],{"className":20463},[437,1313],[17,20465,20467,20481],{"className":20466},[441],[17,20468,20470],{"className":20469,"style":20128},[445],[17,20471,20472,20475],{"style":20131},[17,20473],{"className":20474,"style":20135},[452],[17,20476,20477],{"style":20138},[1332,20478,20479],{"xmlns":1334,"width":1335,"height":20141,"viewBox":20142},[1339,20480],{"d":20145},[17,20482,1346],{"className":20483},[1345],[17,20485,20487],{"className":20486},[441],[17,20488,20490],{"className":20489,"style":20155},[445],[17,20491],{},", selecting the first two rows yields six 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\\begin{vmatrix}   1& 2\\\\  0 &-1 \\end{vmatrix}=-1, \\quad M_{2}= \\begin{vmatrix}  1 &1 \\\\   0& 2\\end{vmatrix}=2, \\quad M_{3}= \\begin{vmatrix} 1  & 4\\\\  0 &1 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\\begin{vmatrix}   2& 1\\\\   -1&2 \\end{vmatrix}=5, \\quad M_{5}= \\begin{vmatrix}   2& 4\\\\  -1 &1 \\end{vmatrix}=6, \\quad M_{6}= \\begin{vmatrix}   1& 4\\\\   2&1 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algebraic cofactors 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\\quad A_2=3, \\quad A_3=-1, \\quad A_4=1, \\quad A_5=-3, \\quad A_6=1 \\Rightarrow D=-7.",[17,22576,22578,22633,22703,22770,22840,22907,22977,22995,23013],{"className":22577,"ariaHidden":48},[47],[17,22579,22581,22584,22624,22627,22630],{"className":22580},[52],[17,22582],{"className":22583,"style":14691},[56],[17,22585,22587,22590],{"className":22586},[61],[17,22588,4935],{"className":22589},[61,62],[17,22591,22593],{"className":22592},[433],[17,22594,22596,22616],{"className":22595},[437,1313],[17,22597,22599,22613],{"className":22598},[441],[17,22600,22602],{"className":22601,"style":1404},[445],[17,22603,22604,22607],{"style":1407},[17,22605],{"className":22606,"style":453},[452],[17,22608,22610],{"className":22609},[457,458,459,460],[17,22611,3544],{"className":22612},[61,460],[17,22614,1346],{"className":22615},[1345],[17,22617,22619],{"className":22618},[441],[17,22620,22622],{"className":22621,"style":1429},[445],[17,22623],{},[17,22625],{"className":22626,"style":372},[228],[17,22628,825],{"className":22629},[376],[17,22631],{"className":22632,"style":372},[228],[17,22634,22636,22639,22642,22645,22648,22651,22654,22694,22697,22700],{"className":22635},[52],[17,22637],{"className":22638,"style":18038},[56],[17,22640,627],{"className":22641},[61],[17,22643,22486],{"className":22644},[61],[17,22646,1084],{"className":22647},[2410],[17,22649],{"className":22650,"style":20947},[228],[17,22652],{"className":22653,"style":2406},[228],[17,22655,22657,22660],{"className":22656},[61],[17,22658,4935],{"className":22659},[61,62],[17,22661,22663],{"className":22662},[433],[17,22664,22666,22686],{"className":22665},[437,1313],[17,22667,22669,22683],{"className":22668},[441],[17,22670,22672],{"className":22671,"style":1404},[445],[17,22673,22674,22677],{"style":1407},[17,22675],{"className":22676,"style":453},[452],[17,22678,22680],{"className":22679},[457,458,459,460],[17,22681,410],{"className":22682},[61,460],[17,22684,1346],{"className":22685},[1345],[17,22687,22689],{"className":22688},[441],[17,22690,22692],{"className":22691,"style":1429},[445],[17,22693],{},[17,22695],{"className":22696,"style":372},[228],[17,22698,825],{"className":22699},[376],[17,22701],{"className":22702,"style":372},[228],[17,22704,22706,22709,22712,22715,22718,22721,22761,22764,22767],{"className":22705},[52],[17,22707],{"className":22708,"style":18038},[56],[17,22710,3577],{"className":22711},[61],[17,22713,1084],{"className":22714},[2410],[17,22716],{"className":22717,"style":20947},[228],[17,22719],{"className":22720,"style":2406},[228],[17,22722,22724,22727],{"className":22723},[61],[17,22725,4935],{"className":22726},[61,62],[17,22728,22730],{"className":22729},[433],[17,22731,22733,22753],{"className":22732},[437,1313],[17,22734,22736,22750],{"className":22735},[441],[17,22737,22739],{"className":22738,"style":1404},[445],[17,22740,22741,22744],{"style":1407},[17,22742],{"className":22743,"style":453},[452],[17,22745,22747],{"className":22746},[457,458,459,460],[17,22748,3577],{"className":22749},[61,460],[17,22751,1346],{"className":22752},[1345],[17,22754,22756],{"className":22755},[441],[17,22757,22759],{"className":22758,"style":1429},[445],[17,22760],{},[17,22762],{"className":22763,"style":372},[228],[17,22765,825],{"className":22766},[376],[17,22768],{"className":22769,"style":372},[228],[17,22771,22773,22776,22779,22782,22785,22788,22791,22831,22834,22837],{"className":22772},[52],[17,22774],{"className":22775,"style":18038},[56],[17,22777,627],{"className":22778},[61],[17,22780,3544],{"className":22781},[61],[17,22783,1084],{"className":22784},[2410],[17,22786],{"className":22787,"style":20947},[228],[17,22789],{"className":22790,"style":2406},[228],[17,22792,22794,22797],{"className":22793},[61],[17,22795,4935],{"className":22796},[61,62],[17,22798,22800],{"className":22799},[433],[17,22801,22803,22823],{"className":22802},[437,1313],[17,22804,22806,22820],{"className":22805},[441],[17,22807,22809],{"className":22808,"style":1404},[445],[17,22810,22811,22814],{"style":1407},[17,22812],{"className":22813,"style":453},[452],[17,22815,22817],{"className":22816},[457,458,459,460],[17,22818,3564],{"className":22819},[61,460],[17,22821,1346],{"className":22822},[1345],[17,22824,22826],{"className":22825},[441],[17,22827,22829],{"className":22828,"style":1429},[445],[17,22830],{},[17,22832],{"className":22833,"style":372},[228],[17,22835,825],{"className":22836},[376],[17,22838],{"className":22839,"style":372},[228],[17,22841,22843,22846,22849,22852,22855,22858,22898,22901,22904],{"className":22842},[52],[17,22844],{"className":22845,"style":18038},[56],[17,22847,3544],{"className":22848},[61],[17,22850,1084],{"className":22851},[2410],[17,22853],{"className":22854,"style":20947},[228],[17,22856],{"className":22857,"style":2406},[228],[17,22859,22861,22864],{"className":22860},[61],[17,22862,4935],{"className":22863},[61,62],[17,22865,22867],{"className":22866},[433],[17,22868,22870,22890],{"className":22869},[437,1313],[17,22871,22873,22887],{"className":22872},[441],[17,22874,22876],{"className":22875,"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 a_{11} & a_{12} & \\cdots &a_{1n} \\\\ a_{21}  &a_{22}  &\\cdots  & a_{2n}\\\\ \\vdots  & \\vdots &  &\\vdots \\\\  a_{n1} & a_{n2} &\\cdots  & a_{nn}\\end{vmatrix}, \\\\ D_{2}=\\begin{vmatrix}  b_{11} & b_{12} & \\cdots &b_{1n} \\\\ b_{21}  &b_{22}  &\\cdots  & b_{2n}\\\\ \\vdots  & \\vdots &  &\\vdots \\\\  b_{n1} & b_{n2} &\\cdots  & 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determinant",[17,25131,25133],{"className":25132,"translate":21},[807],[17,25134,25136,25339],{"className":25135,"translate":21},[20],[17,25137,25139],{"className":25138},[25],[27,25140,25141],{"xmlns":29,"display":816},[31,25142,25143,25336],{},[34,25144,25145,25147,25149],{},[37,25146,18797],{},[204,25148,825],{},[34,25150,25151,25153,25334],{},[204,25152,830],{"fence":48},[832,25154,25155,25198,25240,25284],{"rowspacing":834,"columnalign":19968,"columnspacing":836},[838,25156,25157,25168,25178,25184],{},[841,25158,25159],{},[844,25160,25161],{"scriptlevel":846,"displaystyle":621},[848,25162,25163,25166],{},[37,25164,25165],{},"c",[408,25167,855],{},[841,25169,25170],{},[844,25171,25172],{"scriptlevel":846,"displaystyle":621},[848,25173,25174,25176],{},[37,25175,25165],{},[408,25177,866],{},[841,25179,25180],{},[844,25181,25182],{"scriptlevel":846,"displaystyle":621},[204,25183,3905],{"lspace":694,"rspace":694},[841,25185,25186],{},[844,25187,25188],{"scriptlevel":846,"displaystyle":621},[848,25189,25190,25192],{},[37,25191,25165],{},[34,25193,25194,25196],{},[408,25195,3544],{},[37,25197,80],{},[838,25199,25200,25210,25220,25226],{},[841,25201,25202],{},[844,25203,25204],{"scriptlevel":846,"displaystyle":621},[848,25205,25206,25208],{},[37,25207,25165],{},[408,25209,912],{},[841,25211,25212],{},[844,25213,25214],{"scriptlevel":846,"displaystyle":621},[848,25215,25216,25218],{},[37,25217,25165],{},[408,25219,923],{},[841,25221,25222],{},[844,25223,25224],{"scriptlevel":846,"displaystyle":621},[204,25225,3905],{"lspace":694,"rspace":694},[841,25227,25228],{},[844,25229,25230],{"scriptlevel":846,"displaystyle":621},[848,25231,25232,25234],{},[37,25233,25165],{},[34,25235,25236,25238],{},[408,25237,410],{},[37,25239,80],{},[838,25241,25242,25254,25266,25272],{},[841,25243,25244],{},[844,25245,25246],{"scriptlevel":846,"displaystyle":621},[34,25247,25248,25250],{},[37,25249,967],{"mathvariant":693},[969,25251,25252],{"height":694,"voffset":694},[228,25253],{"mathbackground":973,"width":694,"height":974},[841,25255,25256],{},[844,25257,25258],{"scriptlevel":846,"displaystyle":621},[34,25259,25260,25262],{},[37,25261,967],{"mathvariant":693},[969,25263,25264],{"height":694,"voffset":694},[228,25265],{"mathbackground":973,"width":694,"height":974},[841,25267,25268],{},[844,25269,25270],{"scriptlevel":846,"displaystyle":621},[34,25271],{},[841,25273,25274],{},[844,25275,25276],{"scriptlevel":846,"displaystyle":621},[34,25277,25278,25280],{},[37,25279,967],{"mathvariant":693},[969,25281,25282],{"height":694,"voffset":694},[228,25283],{"mathbackground":973,"width":694,"height":974},[838,25285,25286,25300,25314,25320],{},[841,25287,25288],{},[844,25289,25290],{"scriptlevel":846,"displaystyle":621},[848,25291,25292,25294],{},[37,25293,25165],{},[34,25295,25296,25298],{},[37,25297,80],{},[408,25299,3544],{},[841,25301,25302],{},[844,25303,25304],{"scriptlevel":846,"displaystyle":621},[848,25305,25306,25308],{},[37,25307,25165],{},[34,25309,25310,25312],{},[37,25311,80],{},[408,25313,410],{},[841,25315,25316],{},[844,25317,25318],{"scriptlevel":846,"displaystyle":621},[204,25319,3905],{"lspace":694,"rspace":694},[841,25321,25322],{},[844,25323,25324],{"scriptlevel":846,"displaystyle":621},[848,25325,25326,25328],{},[37,25327,25165],{},[34,25329,25330,25332],{},[37,25331,80],{},[37,25333,80],{},[204,25335,830],{"fence":48},[41,25337,25338],{"encoding":43},"C = \\begin{vmatrix}  c_{11} &c_{12} & \\cdots &c_{1n} \\\\ c_{21}  &c_{22}  &\\cdots  & c_{2n}\\\\ \\vdots  & \\vdots &  &\\vdots \\\\  c_{n1} & c_{n2} &\\cdots  & c_{nn}\\end{vmatrix}",[17,25340,25342,25361],{"className":25341,"ariaHidden":48},[47],[17,25343,25345,25348,25352,25355,25358],{"className":25344},[52],[17,25346],{"className":25347,"style":123},[56],[17,25349,18797],{"className":25350,"style":25351},[61,62],"margin-right:0.07153em;",[17,25353],{"className":25354,"style":372},[228],[17,25356,825],{"className":25357},[376],[17,25359],{"className":25360,"style":372},[228],[17,25362,25364,25367],{"className":25363},[52],[17,25365],{"className":25366,"style":1297},[56],[17,25368,25370,25407,26086],{"className":25369},[1301],[17,25371,25373],{"className":25372},[649],[17,25374,25376],{"className":25375},[1308,1309],[17,25377,25379,25399],{"className":25378},[437,1313],[17,25380,25382,25396],{"className":25381},[441],[17,25383,25385],{"className":25384,"style":1320},[445],[17,25386,25387,25390],{"style":1323},[17,25388],{"className":25389,"style":1327},[452],[17,25391,25392],{"style":1330},[1332,25393,25394],{"xmlns":1334,"width":1335,"height":1336,"viewBox":1337},[1339,25395],{"d":1341},[17,25397,1346],{"className":25398},[1345],[17,25400,25402],{"className":25401},[441],[17,25403,25405],{"className":25404,"style":1353},[445],[17,25406],{},[17,25408,25410],{"className":25409},[61],[17,25411,25413,25609,25612,25615,25811,25814,25817,25881,25884,25887],{"className":25412},[832],[17,25414,25416],{"className":25415},[1365],[17,25417,25419,25601],{"className":25418},[437,1313],[17,25420,25422,25598],{"className":25421},[441],[17,25423,25425,25476,25527,25544],{"className":25424,"style":1375},[445],[17,25426,25427,25430],{"style":1378},[17,25428],{"className":25429,"style":1382},[452],[17,25431,25433],{"className":25432},[61],[17,25434,25436,25439],{"className":25435},[61],[17,25437,25165],{"className":25438},[61,62],[17,25440,25442],{"className":25441},[433],[17,25443,25445,25468],{"className":25444},[437,1313],[17,25446,25448,25465],{"className":25447},[441],[17,25449,25451],{"className":25450,"style":1404},[445],[17,25452,25453,25456],{"style":1407},[17,25454],{"className":25455,"style":453},[452],[17,25457,25459],{"className":25458},[457,458,459,460],[17,25460,25462],{"className":25461},[61,460],[17,25463,855],{"className":25464},[61,460],[17,25466,1346],{"className":25467},[1345],[17,25469,25471],{"className":25470},[441],[17,25472,25474],{"className":25473,"style":1429},[445],[17,25475],{},[17,25477,25478,25481],{"style":1434},[17,25479],{"className":25480,"style":1382},[452],[17,25482,25484],{"className":25483},[61],[17,25485,25487,25490],{"className":25486},[61],[17,25488,25165],{"className":25489},[61,62],[17,25491,25493],{"className":25492},[433],[17,25494,25496,25519],{"className":25495},[437,1313],[17,25497,25499,25516],{"className":25498},[441],[17,25500,25502],{"className":25501,"style":1404},[445],[17,25503,25504,25507],{"style":1407},[17,25505],{"className":25506,"style":453},[452],[17,25508,25510],{"className":25509},[457,458,459,460],[17,25511,25513],{"className":25512},[61,460],[17,25514,912],{"className":25515},[61,460],[17,25517,1346],{"className":25518},[1345],[17,25520,25522],{"className":25521},[441],[17,25523,25525],{"className":25524,"style":1429},[445],[17,25526],{},[17,25528,25529,25532],{"style":1486},[17,25530],{"className":25531,"style":1382},[452],[17,25533,25535],{"className":25534},[61],[17,25536,25538,25541],{"className":25537},[61],[17,25539,967],{"className":25540},[61],[17,25542],{"className":25543,"style":1503},[61,1502],[17,25545,25546,25549],{"style":1506},[17,25547],{"className":25548,"style":1382},[452],[17,25550,25552],{"className":25551},[61],[17,25553,25555,25558],{"className":25554},[61],[17,25556,25165],{"className":25557},[61,62],[17,25559,25561],{"className":25560},[433],[17,25562,25564,25590],{"className":25563},[437,1313],[17,25565,25567,25587],{"className":25566},[441],[17,25568,25570],{"className":25569,"style":1404},[445],[17,25571,25572,25575],{"style":1407},[17,25573],{"className":25574,"style":453},[452],[17,25576,25578],{"className":25577},[457,458,459,460],[17,25579,25581,25584],{"className":25580},[61,460],[17,25582,80],{"className":25583},[61,62,460],[17,25585,3544],{"className":25586},[61,460],[17,25588,1346],{"className":25589},[1345],[17,25591,25593],{"className":25592},[441],[17,25594,25596],{"className":25595,"style":1429},[445],[17,25597],{},[17,25599,1346],{"className":25600},[1345],[17,25602,25604],{"className":25603},[441],[17,25605,25607],{"className":25606,"style":1565},[445],[17,25608],{},[17,25610],{"className":25611,"style":1572},[1571],[17,25613],{"className":25614,"style":1572},[1571],[17,25616,25618],{"className":25617},[1365],[17,25619,25621,25803],{"className":25620},[437,1313],[17,25622,25624,25800],{"className":25623},[441],[17,25625,25627,25678,25729,25746],{"className":25626,"style":1375},[445],[17,25628,25629,25632],{"style":1378},[17,25630],{"className":25631,"style":1382},[452],[17,25633,25635],{"className":25634},[61],[17,25636,25638,25641],{"className":25637},[61],[17,25639,25165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each element ",[17,26127,26129,26151],{"className":26128,"translate":21},[20],[17,26130,26132],{"className":26131},[25],[27,26133,26134],{"xmlns":29},[31,26135,26136,26148],{},[34,26137,26138],{},[848,26139,26140,26142],{},[37,26141,25165],{},[34,26143,26144,26146],{},[37,26145,3890],{},[37,26147,3928],{},[41,26149,26150],{"encoding":43},"c_{ij}",[17,26152,26154],{"className":26153,"ariaHidden":48},[47],[17,26155,26157,26161],{"className":26156},[52],[17,26158],{"className":26159,"style":26160},[56],"height:0.7167em;vertical-align:-0.2861em;",[17,26162,26164,26167],{"className":26163},[61],[17,26165,25165],{"className":26166},[61,62],[17,26168,26170],{"className":26169},[433],[17,26171,26173,26197],{"className":26172},[437,1313],[17,26174,26176,26194],{"className":26175},[441],[17,26177,26179],{"className":26178,"style":19102},[445],[17,26180,26181,26184],{"style":1407},[17,26182],{"className":26183,"style":453},[452],[17,26185,26187],{"className":26186},[457,458,459,460],[17,26188,26190],{"className":26189},[61,460],[17,26191,26193],{"className":26192,"style":4138},[61,62,460],"ij",[17,26195,1346],{"className":26196},[1345],[17,26198,26200],{"className":26199},[441],[17,26201,26203],{"className":26202,"style":5963},[445],[17,26204],{}," is the sum of the products of elements from the ",[17,26207,26209,26222],{"className":26208,"translate":21},[20],[17,26210,26212],{"className":26211},[25],[27,26213,26214],{"xmlns":29},[31,26215,26216,26220],{},[34,26217,26218],{},[37,26219,3890],{},[41,26221,3890],{"encoding":43},[17,26223,26225],{"className":26224,"ariaHidden":48},[47],[17,26226,26228,26232],{"className":26227},[52],[17,26229],{"className":26230,"style":26231},[56],"height:0.6595em;",[17,26233,3890],{"className":26234},[61,62],"-th row of ",[17,26237,26239,26256],{"className":26238,"translate":21},[20],[17,26240,26242],{"className":26241},[25],[27,26243,26244],{"xmlns":29},[31,26245,26246,26254],{},[34,26247,26248],{},[848,26249,26250,26252],{},[37,26251,111],{},[408,26253,3544],{},[41,26255,14681],{"encoding":43},[17,26257,26259],{"className":26258,"ariaHidden":48},[47],[17,26260,26262,26265],{"className":26261},[52],[17,26263],{"className":26264,"style":14691},[56],[17,26266,26268,26271],{"className":26267},[61],[17,26269,111],{"className":26270,"style":127},[61,62],[17,26272,26274],{"className":26273},[433],[17,26275,26277,26297],{"className":26276},[437,1313],[17,26278,26280,26294],{"className":26279},[441],[17,26281,26283],{"className":26282,"style":1404},[445],[17,26284,26285,26288],{"style":14712},[17,26286],{"className":26287,"style":453},[452],[17,26289,26291],{"className":26290},[457,458,459,460],[17,26292,3544],{"className":26293},[61,460],[17,26295,1346],{"className":26296},[1345],[17,26298,26300],{"className":26299},[441],[17,26301,26303],{"className":26302,"style":1429},[445],[17,26304],{}," and the corresponding elements from the ",[17,26307,26309,26322],{"className":26308,"translate":21},[20],[17,26310,26312],{"className":26311},[25],[27,26313,26314],{"xmlns":29},[31,26315,26316,26320],{},[34,26317,26318],{},[37,26319,3928],{},[41,26321,3928],{"encoding":43},[17,26323,26325],{"className":26324,"ariaHidden":48},[47],[17,26326,26328,26331],{"className":26327},[52],[17,26329],{"className":26330,"style":3965},[56],[17,26332,3928],{"className":26333,"style":4138},[61,62],[17,26335,26337,26355],{"className":26336,"translate":21},[20],[17,26338,26340],{"className":26339},[25],[27,26341,26342],{"xmlns":29},[31,26343,26344,26352],{},[34,26345,26346],{},[848,26347,26348,26350],{},[37,26349,111],{},[408,26351,410],{},[41,26353,26354],{"encoding":43},"D_2",[17,26356,26358],{"className":26357,"ariaHidden":48},[47],[17,26359,26361,26364],{"className":26360},[52],[17,26362],{"className":26363,"style":14691},[56],[17,26365,26367,26370],{"className":26366},[61],[17,26368,111],{"className":26369,"style":127},[61,62],[17,26371,26373],{"className":26372},[433],[17,26374,26376,26396],{"className":26375},[437,1313],[17,26377,26379,26393],{"className":26378},[441],[17,26380,26382],{"className":26381,"style":1404},[445],[17,26383,26384,26387],{"style":14712},[17,26385],{"className":26386,"style":453},[452],[17,26388,26390],{"className":26389},[457,458,459,460],[17,26391,410],{"className":26392},[61,460],[17,26394,1346],{"className":26395},[1345],[17,26397,26399],{"className":26398},[441],[17,26400,26402],{"className":26401,"style":1429},[445],[17,26403],{},[10,26405,26406],{},"i.e.",[17,26408,26410],{"className":26409,"translate":21},[807],[17,26411,26413,26507],{"className":26412,"translate":21},[20],[17,26414,26416],{"className":26415},[25],[27,26417,26418],{"xmlns":29,"display":816},[31,26419,26420,26504],{},[34,26421,26422,26432,26434,26444,26454,26456,26466,26476,26478,26480,26482,26492,26502],{},[848,26423,26424,26426],{},[37,26425,25165],{},[34,26427,26428,26430],{},[37,26429,3890],{},[37,26431,3928],{},[204,26433,825],{},[848,26435,26436,26438],{},[37,26437,852],{},[34,26439,26440,26442],{},[37,26441,3890],{},[408,26443,3544],{},[848,26445,26446,26448],{},[37,26447,23297],{},[34,26449,26450,26452],{},[408,26451,3544],{},[37,26453,3928],{},[204,26455,3557],{},[848,26457,26458,26460],{},[37,26459,852],{},[34,26461,26462,26464],{},[37,26463,3890],{},[408,26465,410],{},[848,26467,26468,26470],{},[37,26469,23297],{},[34,26471,26472,26474],{},[408,26473,410],{},[37,26475,3928],{},[204,26477,3557],{},[204,26479,3905],{},[204,26481,3557],{},[848,26483,26484,26486],{},[37,26485,852],{},[34,26487,26488,26490],{},[37,26489,3890],{},[37,26491,80],{},[848,26493,26494,26496],{},[37,26495,23297],{},[34,26497,26498,26500],{},[37,26499,80],{},[37,26501,3928],{},[37,26503,798],{"mathvariant":693},[41,26505,26506],{"encoding":43},"c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \\cdots + a_{in}b_{nj}.",[17,26508,26510,26568,26676,26783,26801],{"className":26509,"ariaHidden":48},[47],[17,26511,26513,26516,26559,26562,26565],{"className":26512},[52],[17,26514],{"className":26515,"style":26160},[56],[17,26517,26519,26522],{"className":26518},[61],[17,26520,25165],{"className":26521},[61,62],[17,26523,26525],{"className":26524},[433],[17,26526,26528,26551],{"className":26527},[437,1313],[17,26529,26531,26548],{"className":26530},[441],[17,26532,26534],{"className":26533,"style":19102},[445],[17,26535,26536,26539],{"style":1407},[17,26537],{"className":26538,"style":453},[452],[17,26540,26542],{"className":26541},[457,458,459,460],[17,26543,26545],{"className":26544},[61,460],[17,26546,26193],{"className":26547,"style":4138},[61,62,460],[17,26549,1346],{"className":26550},[1345],[17,26552,26554],{"className":26553},[441],[17,26555,26557],{"className":26556,"style":5963},[445],[17,26558],{},[17,26560],{"className":26561,"style":372},[228],[17,26563,825],{"className":26564},[376],[17,26566],{"className":26567,"style":372},[228],[17,26569,26571,26575,26621,26667,26670,26673],{"className":26570},[52],[17,26572],{"className":26573,"style":26574},[56],"height:0.9805em;vertical-align:-0.2861em;",[17,26576,26578,26581],{"className":26577},[61],[17,26579,852],{"className":26580},[61,62],[17,26582,26584],{"className":26583},[433],[17,26585,26587,26613],{"className":26586},[437,1313],[17,26588,26590,26610],{"className":26589},[441],[17,26591,26593],{"className":26592,"style":19102},[445],[17,26594,26595,26598],{"style":1407},[17,26596],{"className":26597,"style":453},[452],[17,26599,26601],{"className":26600},[457,458,459,460],[17,26602,26604,26607],{"className":26603},[61,460],[17,26605,3890],{"className":26606},[61,62,460],[17,26608,3544],{"className":26609},[61,460],[17,26611,1346],{"className":26612},[1345],[17,26614,26616],{"className":26615},[441],[17,26617,26619],{"className":26618,"style":1429},[445],[17,26620],{},[17,26622,26624,26627],{"className":26623},[61],[17,26625,23297],{"className":26626},[61,62],[17,26628,26630],{"className":26629},[433],[17,26631,26633,26659],{"className":26632},[437,1313],[17,26634,26636,26656],{"className":26635},[441],[17,26637,26639],{"className":26638,"style":19102},[445],[17,26640,26641,26644],{"style":1407},[17,26642],{"className":26643,"style":453},[452],[17,26645,26647],{"className":26646},[457,458,459,460],[17,26648,26650,26653],{"className":26649},[61,460],[17,26651,3544],{"className":26652},[61,460],[17,26654,3928],{"className":26655,"style":4138},[61,62,460],[17,26657,1346],{"className":26658},[1345],[17,26660,26662],{"className":26661},[441],[17,26663,26665],{"className":26664,"style":5963},[445],[17,26666],{},[17,26668],{"className":26669,"style":229},[228],[17,26671,3557],{"className":26672},[233],[17,26674],{"className":26675,"style":229},[228],[17,26677,26679,26682,26728,26774,26777,26780],{"className":26678},[52],[17,26680],{"className":26681,"style":26574},[56],[17,26683,26685,26688],{"className":26684},[61],[17,26686,852],{"className":26687},[61,62],[17,26689,26691],{"className":26690},[433],[17,26692,26694,26720],{"className":26693},[437,1313],[17,26695,26697,26717],{"className":26696},[441],[17,26698,26700],{"className":26699,"style":19102},[445],[17,26701,26702,26705],{"style":1407},[17,26703],{"className":26704,"style":453},[452],[17,26706,26708],{"className":26707},[457,458,459,460],[17,26709,26711,26714],{"className":26710},[61,460],[17,26712,3890],{"className":26713},[61,62,460],[17,26715,410],{"className":26716},[61,460],[17,26718,1346],{"className":26719},[1345],[17,26721,26723],{"className":26722},[441],[17,26724,26726],{"className":26725,"style":1429},[445],[17,26727],{},[17,26729,26731,26734],{"className":26730},[61],[17,26732,23297],{"className":26733},[61,62],[17,26735,26737],{"className":26736},[433],[17,26738,26740,26766],{"className":26739},[437,1313],[17,26741,26743,26763],{"className":26742},[441],[17,26744,26746],{"className":26745,"style":19102},[445],[17,26747,26748,26751],{"style":1407},[17,26749],{"className":26750,"style":453},[452],[17,26752,26754],{"className":26753},[457,458,459,460],[17,26755,26757,26760],{"className":26756},[61,460],[17,26758,410],{"className":26759},[61,460],[17,26761,3928],{"className":26762,"style":4138},[61,62,460],[17,26764,1346],{"className":26765},[1345],[17,26767,26769],{"className":26768},[441],[17,26770,26772],{"className":26771,"style":5963},[445],[17,26773],{},[17,26775],{"className":26776,"style":229},[228],[17,26778,3557],{"className":26779},[233],[17,26781],{"className":26782,"style":229},[228],[17,26784,26786,26789,26792,26795,26798],{"className":26785},[52],[17,26787],{"className":26788,"style":13282},[56],[17,26790,3905],{"className":26791},[1301],[17,26793],{"className":26794,"style":229},[228],[17,26796,3557],{"className":26797},[233],[17,26799],{"className":26800,"style":229},[228],[17,26802,26804,26807,26851,26895],{"className":26803},[52],[17,26805],{"className":26806,"style":26574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to Laplace's theorem, expanding ",[17,29416,29418,29431],{"className":29417,"translate":21},[20],[17,29419,29421],{"className":29420},[25],[27,29422,29423],{"xmlns":29},[31,29424,29425,29429],{},[34,29426,29427],{},[37,29428,111],{},[41,29430,111],{"encoding":43},[17,29432,29434],{"className":29433,"ariaHidden":48},[47],[17,29435,29437,29440],{"className":29436},[52],[17,29438],{"className":29439,"style":123},[56],[17,29441,111],{"className":29442,"style":127},[61,62]," by the first ",[17,29445,29447,29460],{"className":29446,"translate":21},[20],[17,29448,29450],{"className":29449},[25],[27,29451,29452],{"xmlns":29},[31,29453,29454,29458],{},[34,29455,29456],{},[37,29457,80],{},[41,29459,80],{"encoding":43},[17,29461,29463],{"className":29462,"ariaHidden":48},[47],[17,29464,29466,29469],{"className":29465},[52],[17,29467],{"className":29468,"style":92},[56],[17,29470,80],{"className":29471},[61,62]," rows, all of the minors formed by the first ",[17,29474,29476,29489],{"className":29475,"translate":21},[20],[17,29477,29479],{"className":29478},[25],[27,29480,29481],{"xmlns":29},[31,29482,29483,29487],{},[34,29484,29485],{},[37,29486,80],{},[41,29488,80],{"encoding":43},[17,29490,29492],{"className":29491,"ariaHidden":48},[47],[17,29493,29495,29498],{"className":29494},[52],[17,29496],{"className":29497,"style":92},[56],[17,29499,80],{"className":29500},[61,62]," rows and other elements will be zero except for the minor at the top left, so",[17,29503,29505],{"className":29504,"translate":21},[807],[17,29506,29508,29915],{"className":29507,"translate":21},[20],[17,29509,29511],{"className":29510},[25],[27,29512,29513],{"xmlns":29,"display":816},[31,29514,29515,29912],{},[34,29516,29517,29519,29521,29707,29710,29896,29898,29904,29910],{},[37,29518,111],{},[204,29520,825],{},[34,29522,29523,29525,29705],{},[204,29524,830],{"fence":48},[832,29526,29527,29569,29611,29655],{"rowspacing":834,"columnalign":19968,"columnspacing":836},[838,29528,29529,29539,29549,29555],{},[841,29530,29531],{},[844,29532,29533],{"scriptlevel":846,"displaystyle":621},[848,29534,29535,29537],{},[37,29536,852],{},[408,29538,855],{},[841,29540,29541],{},[844,29542,29543],{"scriptlevel":846,"displaystyle":621},[848,29544,29545,29547],{},[37,29546,852],{},[408,29548,866],{},[841,29550,29551],{},[844,29552,29553],{"scriptlevel":846,"displaystyle":621},[204,29554,3905],{"lspace":694,"rspace":694},[841,29556,29557],{},[844,29558,29559],{"scriptlevel":846,"displaystyle":621},[848,29560,29561,29563],{},[37,29562,852],{},[34,29564,29565,29567],{},[408,29566,3544],{},[37,29568,80],{},[838,29570,29571,29581,29591,29597],{},[841,29572,29573],{},[844,29574,29575],{"scriptlevel":846,"displaystyle":621},[848,29576,29577,29579],{},[37,29578,852],{},[408,29580,912],{},[841,29582,29583],{},[844,29584,29585],{"scriptlevel":846,"displaystyle":621},[848,29586,29587,29589],{},[37,29588,852],{},[408,29590,923],{},[841,29592,29593],{},[844,29594,29595],{"scriptlevel":846,"displaystyle":621},[204,29596,3905],{"lspace":694,"rspace":694},[841,29598,29599],{},[844,29600,29601],{"scriptlevel":846,"displaystyle":621},[848,29602,29603,29605],{},[37,29604,852],{},[34,29606,29607,29609],{},[408,29608,410],{},[37,29610,80],{},[838,29612,29613,29625,29637,29643],{},[841,29614,29615],{},[844,29616,29617],{"scriptlevel":846,"displaystyle":621},[34,29618,29619,29621],{},[37,29620,967],{"mathvariant":693},[969,29622,29623],{"height":694,"voffset":694},[228,29624],{"mathbackground":973,"width":694,"height":974},[841,29626,29627],{},[844,29628,29629],{"scriptlevel":846,"displaystyle":621},[34,29630,29631,29633],{},[37,29632,967],{"mathvariant":693},[969,29634,29635],{"height":694,"voffset":694},[228,29636],{"mathbackground":973,"width":694,"height":974},[841,29638,29639],{},[844,29640,29641],{"scriptlevel":846,"displaystyle":621},[34,29642],{},[841,29644,29645],{},[844,29646,29647],{"scriptlevel":846,"displaystyle":621},[34,29648,29649,29651],{},[37,29650,967],{"mathvariant":693},[969,29652,29653],{"height":694,"voffset":694},[228,29654],{"mathbackground":973,"width":694,"height":974},[838,29656,29657,29671,29685,29691],{},[841,29658,29659],{},[844,29660,29661],{"scriptlevel":846,"displaystyle":621},[848,29662,29663,29665],{},[37,29664,852],{},[34,29666,29667,29669],{},[37,29668,80],{},[408,29670,3544],{},[841,29672,29673],{},[844,29674,29675],{"scriptlevel":846,"displaystyle":621},[848,29676,29677,29679],{},[37,29678,852],{},[34,29680,29681,29683],{},[37,29682,80],{},[408,29684,410],{},[841,29686,29687],{},[844,29688,29689],{"scriptlevel":846,"displaystyle":621},[204,29690,3905],{"lspace":694,"rspace":694},[841,29692,29693],{},[844,29694,29695],{"scriptlevel":846,"displaystyle":621},[848,29696,29697,29699],{},[37,29698,852],{},[34,29700,29701,29703],{},[37,29702,80],{},[37,29704,80],{},[204,29706,830],{"fence":48},[204,29708,29709],{},"⋅",[34,29711,29712,29714,29894],{},[204,29713,830],{"fence":48},[832,29715,29716,29758,29800,29844],{"rowspacing":834,"columnalign":19968,"columnspacing":836},[838,29717,29718,29728,29738,29744],{},[841,29719,29720],{},[844,29721,29722],{"scriptlevel":846,"displaystyle":621},[848,29723,29724,29726],{},[37,29725,23297],{},[408,29727,855],{},[841,29729,29730],{},[844,29731,29732],{"scriptlevel":846,"displaystyle":621},[848,29733,29734,29736],{},[37,29735,23297],{},[408,29737,866],{},[841,29739,29740],{},[844,29741,29742],{"scriptlevel":846,"displaystyle":621},[204,29743,3905],{"lspace":694,"rspace":694},[841,29745,29746],{},[844,29747,29748],{"scriptlevel":846,"displaystyle":621},[848,29749,29750,29752],{},[37,29751,23297],{},[34,29753,29754,29756],{},[408,29755,3544],{},[37,29757,80],{},[838,29759,29760,29770,29780,29786],{},[841,29761,29762],{},[844,29763,29764],{"scriptlevel":846,"displaystyle":621},[848,29765,29766,29768],{},[37,29767,23297],{},[408,29769,912],{},[841,29771,29772],{},[844,29773,29774],{"scriptlevel":846,"displaystyle":621},[848,29775,29776,29778],{},[37,29777,23297],{},[408,29779,923],{},[841,29781,29782],{},[844,29783,29784],{"scriptlevel":846,"displaystyle":621},[204,29785,3905],{"lspace":694,"rspace":694},[841,29787,29788],{},[844,29789,29790],{"scriptlevel":846,"displaystyle":621},[848,29791,29792,29794],{},[37,29793,23297],{},[34,29795,29796,29798],{},[408,29797,410],{},[37,29799,80],{},[838,29801,29802,29814,29826,29832],{},[841,29803,29804],{},[844,29805,29806],{"scriptlevel":846,"displaystyle":621},[34,29807,29808,29810],{},[37,29809,967],{"mathvaria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= \\begin{vmatrix}  a_{11} & a_{12} & \\cdots &a_{1n} \\\\ a_{21}  &a_{22}  &\\cdots  & a_{2n}\\\\ \\vdots  & \\vdots &  &\\vdots \\\\  a_{n1} & a_{n2} &\\cdots  & a_{nn}\\end{vmatrix}\\cdot \\begin{vmatrix}  b_{11} & b_{12} & \\cdots &b_{1n} \\\\ b_{21}  &b_{22}  &\\cdots  & b_{2n}\\\\ \\vdots  & \\vdots &  &\\vdots \\\\  b_{n1} & b_{n2} &\\cdots  & 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we show that ",[17,31571,31573,31591],{"className":31572,"translate":21},[20],[17,31574,31576],{"className":31575},[25],[27,31577,31578],{"xmlns":29},[31,31579,31580,31588],{},[34,31581,31582,31584,31586],{},[37,31583,111],{},[204,31585,825],{},[37,31587,18797],{},[41,31589,31590],{"encoding":43},"D = C",[17,31592,31594,31612],{"className":31593,"ariaHidden":48},[47],[17,31595,31597,31600,31603,31606,31609],{"className":31596},[52],[17,31598],{"className":31599,"style":123},[56],[17,31601,111],{"className":31602,"style":127},[61,62],[17,31604],{"className":31605,"style":372},[228],[17,31607,825],{"className":31608},[376],[17,31610],{"className":31611,"style":372},[228],[17,31613,31615,31618],{"className":31614},[52],[17,31616],{"className":31617,"style":123},[56],[17,31619,18797],{"className":31620,"style":25351},[61,62],". Perform elementary row operations on ",[17,31623,31625,31638],{"className":31624,"translate":21},[20],[17,31626,31628],{"className":31627},[25],[27,31629,31630],{"xmlns":29},[31,31631,31632,31636],{},[34,31633,31634],{},[37,31635,111],{},[41,31637,111],{"encoding":43},[17,31639,31641],{"className":31640,"ariaHidden":48},[47],[17,31642,31644,31647],{"className":31643},[52],[17,31645],{"className":31646,"style":123},[56],[17,31648,111],{"className":31649,"style":127},[61,62],": add the ",[17,31652,31654,31672],{"className":31653,"translate":21},[20],[17,31655,31657],{"className":31656},[25],[27,31658,31659],{"xmlns":29},[31,31660,31661,31669],{},[34,31662,31663],{},[848,31664,31665,31667],{},[37,31666,852],{},[408,31668,855],{},[41,31670,31671],{"encoding":43},"a_{11}",[17,31673,31675],{"className":31674,"ariaHidden":48},[47],[17,31676,31678,31681],{"className":31677},[52],[17,31679],{"className":31680,"style":8398},[56],[17,31682,31684,31687],{"className":31683},[61],[17,31685,852],{"className":31686},[61,62],[17,31688,31690],{"className":31689},[433],[17,31691,31693,31716],{"className":31692},[437,1313],[17,31694,31696,31713],{"className":31695},[441],[17,31697,31699],{"className":31698,"style":1404},[445],[17,31700,31701,31704],{"style":1407},[17,31702],{"className":31703,"style":453},[452],[17,31705,31707],{"className":31706},[457,458,459,460],[17,31708,31710],{"className":31709},[61,460],[17,31711,855],{"className":31712},[61,460],[17,31714,1346],{"className":31715},[1345],[17,31717,31719],{"className":31718},[441],[17,31720,31722],{"className":31721,"style":1429},[445],[17,31723],{},"-multiple of the ",[17,31726,31728,31750],{"className":31727,"translate":21},[20],[17,31729,31731],{"className":31730},[25],[27,31732,31733],{"xmlns":29},[31,31734,31735,31747],{},[34,31736,31737,31739,31741,31743,31745],{},[204,31738,622],{"stretchy":621},[37,31740,80],{},[204,31742,3557],{},[408,31744,3544],{},[204,31746,632],{"stretchy":621},[41,31748,31749],{"encoding":43},"(n+1)",[17,31751,31753,31774],{"className":31752,"ariaHidden":48},[47],[17,31754,31756,31759,31762,31765,31768,31771],{"className":31755},[52],[17,31757],{"className":31758,"style":645},[56],[17,31760,622],{"className":31761},[649],[17,31763,80],{"className":31764},[61,62],[17,31766],{"className":31767,"style":229},[228],[17,31769,3557],{"className":31770},[233],[17,31772],{"className":31773,"style":229},[228],[17,31775,31777,31780,31783],{"className":31776},[52],[17,31778],{"className":31779,"style":645},[56],[17,31781,3544],{"className":31782},[61],[17,31784,632],{"className":31785},[674],"-th row, the ",[17,31788,31790,31808],{"className":31789,"translate":21},[20],[17,31791,31793],{"className":31792},[25],[27,31794,31795],{"xmlns":29},[31,31796,31797,31805],{},[34,31798,31799],{},[848,31800,31801,31803],{},[37,31802,852],{},[408,31804,866],{},[41,31806,31807],{"encoding":43},"a_{12}",[17,31809,31811],{"className":31810,"ariaHidden":48},[47],[17,31812,31814,31817],{"className":31813},[52],[17,31815],{"className":31816,"style":8398},[56],[17,31818,31820,31823],{"className":31819},[61],[17,31821,852],{"className":31822},[61,62],[17,31824,31826],{"className":31825},[433],[17,31827,31829,31852],{"className":31828},[437,1313],[17,31830,31832,31849],{"className":31831},[441],[17,31833,31835],{"className":31834,"style":1404},[445],[17,31836,31837,31840],{"style":1407},[17,31838],{"className":31839,"style":453},[452],[17,31841,31843],{"className":31842},[457,458,459,460],[17,31844,31846],{"className":31845},[61,460],[17,31847,866],{"className":31848},[61,460],[17,31850,1346],{"className":31851},[1345],[17,31853,31855],{"className":31854},[441],[17,31856,31858],{"className":31857,"style":1429},[445],[17,31859],{},[17,31861,31863,31885],{"className":31862,"translate":21},[20],[17,31864,31866],{"className":31865},[25],[27,31867,31868],{"xmlns":29},[31,31869,31870,31882],{},[34,31871,31872,31874,31876,31878,31880],{},[204,31873,622],{"stretchy":621},[37,31875,80],{},[204,31877,3557],{},[408,31879,410],{},[204,31881,632],{"stretchy":621},[41,31883,31884],{"encoding":43},"(n+2)",[17,31886,31888,31909],{"className":31887,"ariaHidden":48},[47],[17,31889,31891,31894,31897,31900,31903,31906],{"className":31890},[52],[17,31892],{"className":31893,"style":645},[56],[17,31895,622],{"className":31896},[649],[17,31898,80],{"className":31899},[61,62],[17,31901],{"className":31902,"style":229},[228],[17,31904,3557],{"className":31905},[233],[17,31907],{"className":31908,"style":229},[228],[17,31910,31912,31915,31918],{"className":31911},[52],[17,31913],{"className":31914,"style":645},[56],[17,31916,410],{"className":31917},[61],[17,31919,632],{"className":31920},[674],"-th row, ",[17,31923,31925,31939],{"className":31924,"translate":21},[20],[17,31926,31928],{"className":31927},[25],[27,31929,31930],{"xmlns":29},[31,31931,31932,31936],{},[34,31933,31934],{},[204,31935,27331],{},[41,31937,31938],{"encoding":43},"\\dots",[17,31940,31942],{"className":31941,"ariaHidden":48},[47],[17,31943,31945,31949],{"className":31944},[52],[17,31946],{"className":31947,"style":31948},[56],"height:0.123em;",[17,31950,27331],{"className":31951},[1301],", and the ",[17,31954,31956,31978],{"className":31955,"translate":21},[20],[17,31957,31959],{"className":31958},[25],[27,31960,31961],{"xmlns":29},[31,31962,31963,31975],{},[34,31964,31965],{},[848,31966,31967,31969],{},[37,31968,852],{},[34,31970,31971,31973],{},[408,31972,3544],{},[37,31974,80],{},[41,31976,31977],{"encoding":43},"a_{1n}",[17,31979,31981],{"className":31980,"ariaHidden":48},[47],[17,31982,31984,31987],{"className":31983},[52],[17,31985],{"className":31986,"style":8398},[56],[17,31988,31990,31993],{"className":31989},[61],[17,31991,852],{"className":31992},[61,62],[17,31994,31996],{"className":31995},[433],[17,31997,31999,32025],{"className":31998},[437,1313],[17,32000,32002,32022],{"className":32001},[441],[17,32003,32005],{"className":32004,"style":1404},[445],[17,32006,32007,32010],{"style":1407},[17,32008],{"className":32009,"style":453},[452],[17,32011,32013],{"className":32012},[457,458,459,460],[17,32014,32016,32019],{"className":32015},[61,460],[17,32017,3544],{"className":32018},[61,460],[17,32020,80],{"className":32021},[61,62,460],[17,32023,1346],{"className":32024},[1345],[17,32026,32028],{"className":32027},[441],[17,32029,32031],{"className":32030,"style":1429},[445],[17,32032],{},[17,32034,32036,32056],{"className":32035,"translate":21},[20],[17,32037,32039],{"className":32038},[25],[27,32040,32041],{"xmlns":29},[31,32042,32043,32053],{},[34,32044,32045,32047,32049,32051],{},[204,32046,622],{"stretchy":621},[408,32048,410],{},[37,32050,80],{},[204,32052,632],{"stretchy":621},[41,32054,32055],{"encoding":43},"(2n)",[17,32057,32059],{"className":32058,"ariaHidden":48},[47],[17,32060,32062,32065,32068,32071,32074],{"className":32061},[52],[17,32063],{"className":32064,"style":645},[56],[17,32066,622],{"className":32067},[649],[17,32069,410],{"className":32070},[61],[17,32072,80],{"className":32073},[61,62],[17,32075,632],{"className":32076},[674],"-th row to the first row to get",[17,32079,32081],{"className":32080,"translate":21},[807],[17,32082,32084,32688],{"className":32083,"translate":21},[20],[17,32085,32087],{"className":32086},[25],[27,32088,32089],{"xmlns":29,"display":816},[31,32090,32091,32685],{},[34,32092,32093,32095,32097,32683],{},[37,32094,111],{},[204,32096,825],{},[34,32098,32099,32101,32681],{},[204,32100,830],{"fence":48},[832,32102,32103,32169,32235,32303,32377,32447,32517,32603],{"rowspacing":834,"columnalign":5063,"columnspacing":836},[838,32104,32105,32111,32117,32123,32129,32139,32149,32155],{},[841,32106,32107],{},[844,32108,32109],{"scriptlevel":846,"displaystyle":621},[408,32110,846],{},[841,32112,32113],{},[844,32114,32115],{"scriptlevel":846,"displaystyle":621},[408,32116,846],{},[841,32118,32119],{},[844,32120,32121],{"scriptlevel":846,"displaystyle":621},[204,32122,3905],{"lspace":694,"rspace":694},[841,32124,32125],{},[844,32126,32127],{"scriptlevel":846,"displaystyle":621},[408,32128,846],{},[841,32130,32131],{},[844,32132,32133],{"scriptlevel":846,"displaystyle":621},[848,32134,32135,32137],{},[37,32136,25165],{},[408,32138,855],{},[841,32140,32141],{},[844,32142,32143],{"scriptlevel":846,"displaystyle":621},[848,32144,32145,32147],{},[37,32146,25165],{},[408,32148,866],{},[841,32150,32151],{},[844,32152,32153],{"scriptlevel":846,"displaystyle":621},[204,32154,3905],{"lspace":694,"rspace":694},[841,32156,32157],{},[844,32158,32159],{"scriptlevel":846,"displaystyle":621},[848,32160,32161,32163],{},[37,32162,25165],{},[34,32164,32165,32167],{},[408,32166,3544],{},[37,32168,80],{},[838,32170,32171,32181,32191,32197,32211,32217,32223,32229],{},[841,32172,32173],{},[844,32174,32175],{"scriptlevel":846,"displaystyle":621},[848,32176,32177,32179],{},[37,32178,852],{},[408,32180,912],{},[841,32182,32183],{},[844,32184,32185],{"scriptlevel":846,"displaystyle":621},[848,32186,32187,32189],{},[37,32188,852],{},[408,32190,923],{},[841,32192,32193],{},[844,32194,32195],{"scriptlevel":846,"displaystyle":621},[204,32196,3905],{"lspace":694,"rspace":694},[841,32198,32199],{},[844,32200,32201],{"scriptlevel":846,"displaystyle":621},[848,32202,32203,32205],{},[37,32204,852],{},[34,32206,32207,32209],{},[408,32208,410],{},[37,32210,80],{},[841,32212,32213],{},[844,32214,32215],{"scriptlevel":846,"displaystyle":621},[408,32216,846],{},[841,32218,32219],{},[844,32220,32221],{"scriptlevel":846,"displaystyle":621},[408,32222,846],{},[841,32224,32225],{},[844,32226,32227],{"scriptlevel":846,"displaystyle":621},[408,32228,846],{},[841,32230,32231],{},[844,32232,32233],{"scriptlevel":846,"displaystyle":621},[408,32234,846],{},[838,32236,32237,32249,32261,32267,32279,32285,32291,32297],{},[841,32238,32239],{},[844,32240,32241],{"scriptlevel":846,"displaystyle":621},[34,32242,32243,32245],{},[37,32244,967],{"mathvariant":693},[969,32246,32247],{"height":694,"voffset":694},[228,32248],{"mathbackground":973,"width":694,"height":974},[841,32250,32251],{},[844,32252,32253],{"scriptlevel":846,"displaystyle":621},[34,32254,32255,32257],{},[37,32256,967],{"mathvariant":693},[969,32258,32259],{"height":694,"voffset":694},[228,32260],{"mathbackground":973,"width":694,"height":974},[841,32262,32263],{},[844,32264,32265],{"scriptlevel":846,"displaystyle":621},[34,32266],{},[841,32268,32269],{},[844,32270,32271],{"scriptlevel":846,"displaystyle":621},[34,32272,32273,32275],{},[37,32274,967],{"mathvariant":693},[969,32276,32277],{"height":694,"voffset":694},[228,32278],{"mathbackground":973,"width":694,"height":974},[841,32280,32281],{},[844,32282,32283],{"scriptlevel":846,"displaystyle":621},[408,32284,846],{},[841,32286,32287],{},[844,32288,32289],{"scriptlevel":846,"displaystyle":621},[408,32290,846],{},[841,32292,32293],{},[844,32294,32295],{"scriptlevel":846,"displaystyle":621},[408,32296,846],{},[841,32298,32299],{},[844,32300,32301],{"scriptlevel":846,"displaystyle":621},[408,32302,846],{},[838,32304,32305,32319,32333,32339,32353,32359,32365,32371],{},[841,32306,32307],{},[844,32308,32309],{"scriptlevel":846,"displaystyle":621},[848,32310,32311,32313],{},[37,32312,852],{},[34,32314,32315,32317],{},[37,32316,80],{},[408,32318,3544],{},[841,32320,32321],{},[844,32322,32323],{"scriptlevel":846,"displaystyle":621},[848,32324,32325,32327],{},[37,32326,852],{},[34,32328,32329,32331],{},[37,32330,80],{},[408,32332,410],{},[841,32334,32335],{},[844,32336,32337],{"scriptlevel":846,"displaystyle":621},[204,32338,3905],{"lspace":694,"rspace":694},[841,32340,32341],{},[844,32342,32343],{"scriptlevel":846,"displaystyle":621},[848,32344,32345,32347],{},[37,32346,852],{},[34,32348,32349,32351],{},[37,32350,80],{},[37,32352,80],{},[841,32354,32355],{},[844,32356,32357],{"scriptlevel":846,"displaystyle":621},[408,32358,846],{},[841,32360,32361],{},[844,32362,32363],{"scriptlevel":846,"displaystyle":621},[408,32364,846],{},[841,32366,32367],{},[844,32368,32369],{"scriptlevel":846,"displaystyle":621},[408,32370,846],{},[841,32372,32373],{},[844,32374,32375],{"scriptlevel":846,"displaystyle":621},[408,32376,846],{},[838,32378,32379,32389,32395,32401,32407,32417,32427,32433],{},[841,32380,32381],{},[844,32382,32383],{"scriptlevel":846,"displaystyle":621},[34,32384,32385,32387],{},[204,32386,627],{},[408,32388,3544],{},[841,32390,32391],{},[844,32392,32393],{"scriptlevel":846,"displaystyle":621},[408,32394,846],{},[841,32396,32397],{},[844,32398,32399],{"scriptlevel":846,"displaystyle":621},[204,32400,3905],{"lspace":694,"rspace":694},[841,32402,32403],{},[844,32404,32405],{"scriptlevel":846,"displaystyle":621},[408,32406,846],{},[841,32408,32409],{},[844,32410,32411],{"scriptlevel":846,"displaystyle":621},[848,32412,32413,32415],{},[37,32414,23297],{},[408,32416,855],{},[841,32418,32419],{},[844,32420,32421],{"scriptlevel":846,"displaystyle":621},[848,32422,32423,32425],{},[37,32424,23297],{},[408,32426,866],{},[841,32428,32429],{},[844,32430,32431],{"scriptlevel":846,"displaystyle":621},[204,32432,3905],{"lspace":694,"rspace":694},[841,32434,32435],{},[844,32436,32437],{"scriptlevel":846,"displaystyle":621},[848,32438,32439,32441],{},[37,32440,23297],{},[34,32442,32443,32445],{},[408,32444,3544],{},[37,32446,80],{},[838,32448,32449,32455,32465,32471,32477,32487,32497,32503],{},[841,32450,32451],{},[844,32452,32453],{"scriptlevel":846,"displaystyle":621},[408,32454,846],{},[841,32456,32457],{},[844,32458,32459],{"scriptlevel":846,"displaystyle":621},[34,32460,32461,32463],{},[204,32462,627],{},[408,32464,3544],{},[841,32466,32467],{},[844,32468,32469],{"scriptlevel":846,"displaystyle":621},[204,32470,27331],{"lspace":694,"rspace":694},[841,32472,32473],{},[844,32474,32475],{"scriptlevel":846,"displaystyle":621},[408,32476,846],{},[841,32478,32479],{},[844,32480,32481],{"scriptlevel":846,"displaystyle":621},[848,32482,32483,32485],{},[37,32484,23297],{},[408,32486,912],{},[841,32488,32489],{},[844,32490,32491],{"scriptlevel":846,"displaystyle":621},[848,32492,32493,32495],{},[37,32494,23297],{},[408,32496,923],{},[841,32498,32499],{},[844,32500,32501],{"scriptlevel":846,"displaystyle":621},[204,32502,3905],{"lspace":694,"rspace":694},[841,32504,32505],{},[844,32506,32507],{"scriptlevel":846,"displaystyle":621},[848,32508,32509,32511],{},[37,32510,23297],{},[34,32512,32513,32515],{},[408,32514,410],{},[37,32516,80],{},[838,32518,32519,32531,32543,32549,32561,32573,32585,32591],{},[841,32520,32521],{},[844,32522,32523],{"scriptlevel":846,"displaystyle":621},[34,32524,32525,32527],{},[37,32526,967],{"mathvariant":693},[969,32528,32529],{"height":694,"voffset":694},[228,32530],{"mathbackground":973,"width":694,"height":974},[841,32532,32533],{},[844,32534,32535],{"scriptlevel":846,"displaystyle":621},[34,32536,32537,32539],{},[37,32538,967],{"mathvariant":693},[969,32540,32541],{"height":694,"voffset":694},[228,32542],{"mathbackground":973,"width":694,"height":974},[841,32544,32545],{},[844,32546,32547],{"scriptlevel":846,"displaystyle":621},[34,32548],{},[841,32550,32551],{},[844,32552,32553],{"scriptlevel":846,"displaystyle":621},[34,32554,32555,32557],{},[37,32556,967],{"mathvariant":693},[969,32558,32559],{"height":694,"voffset":694},[228,32560],{"mathbackground":973,"width":694,"height":974},[841,32562,32563],{},[844,32564,32565],{"scriptlevel":846,"displaystyle":621},[34,32566,32567,32569],{},[37,32568,967],{"mathvariant":693},[969,32570,32571],{"height":694,"voffset":694},[228,32572],{"mathbackground":973,"width":694,"height":974},[841,32574,32575],{},[844,32576,32577],{"scriptlevel":846,"displaystyle":621},[34,32578,32579,32581],{},[37,32580,967],{"mathvariant":693},[969,32582,32583],{"height":694,"voffset":694},[228,32584],{"mathbackground":973,"width":694,"height":974},[841,32586,32587],{},[844,32588,32589],{"scriptlevel":846,"displaystyle":621},[34,32590],{},[841,32592,32593],{},[844,32594,32595],{"scriptlevel":846,"displaystyle":621},[34,32596,32597,32599],{},[37,32598,967],{"mathvariant":693},[969,32600,32601],{"height":694,"voffset":694},[228,32602],{"mathbackground":973,"width":694,"height":974},[838,32604,32605,32611,32617,32623,32633,32647,32661,32667],{},[841,32606,32607],{},[844,32608,32609],{"scriptlevel":846,"displaystyle":621},[408,32610,846],{},[841,32612,32613],{},[844,32614,32615],{"scriptlevel":846,"displaystyle":621},[408,32616,846],{},[841,32618,32619],{},[844,32620,32621],{"scriptlevel":846,"displaystyle":621},[204,32622,3905],{"lspace":694,"rspace":694},[841,32624,32625],{},[844,32626,32627],{"scriptlevel":846,"displaystyle":621},[34,32628,32629,32631],{},[204,32630,627],{},[408,32632,3544],{},[841,32634,32635],{},[844,32636,32637],{"scriptlevel":846,"displaystyle":621},[848,32638,32639,32641],{},[37,32640,23297],{},[34,32642,32643,32645],{},[37,32644,80],{},[408,32646,3544],{},[841,32648,32649],{},[844,32650,32651],{"scriptlevel":846,"displaystyle":621},[848,32652,32653,32655],{},[37,32654,23297],{},[34,32656,32657,32659],{},[37,32658,80],{},[408,32660,410],{},[841,32662,32663],{},[844,32664,32665],{"scriptlevel":846,"displaystyle":621},[204,32666,3905],{"lspace":694,"rspac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= \\begin{vmatrix}  0&0&\\cdots&0&c_{11}&c_{12}&\\cdots&c_{1n}\\\\ a_{21}  &a_{22}  &\\cdots  & a_{2n}&0&0&0&0\\\\ \\vdots  & \\vdots &  &\\vdots &0&0&0&0\\\\  a_{n1} & a_{n2} &\\cdots  & a_{nn} &0&0&0&0\\\\ -1&0&\\cdots &0&b_{11}&b_{12}&\\cdots&b_{1n}\\\\ 0&-1&\\dotso&0&b_{21}&b_{22}&\\cdots&b_{2n}\\\\ \\vdots & \\vdots & &\\vdots&\\vdots & \\vdots &&\\vdots \\\\ 0&0&\\cdots&-1 &b_{n1}&b_{n2}&\\cdots &b_{nn} 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= \\begin{vmatrix}  0&0&\\cdots&0& a_{11} & a_{12} & \\cdots &a_{1n} \\\\ 0&0&\\cdots&0&a_{21}  &a_{22}  &\\cdots  & a_{2n}\\\\ \\vdots &\\vdots & &\\vdots &\\vdots  & \\vdots &  &\\vdots \\\\  0&0&\\cdots&0&a_{n1} & a_{n2} &\\cdots  & a_{nn} \\\\ -1&0&\\cdots &0&b_{11}&b_{12}&\\cdots&b_{1n}\\\\ 0&-1&\\dotso&0&b_{21}&b_{22}&\\cdots&b_{2n}\\\\ \\vdots & \\vdots & &\\vdots&\\vdots & \\vdots &&\\vdots \\\\ 0&0&\\cdots&-1 &b_{n1}&b_{n2}&\\cdots &b_{nn} 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1345],[17,37470,37472],{"className":37471},[441],[17,37473,37475],{"className":37474,"style":27620},[445],[17,37476],{},[17,37478],{"className":37479,"style":2406},[228],[17,37481,798],{"className":37482},[61],[10,37484,37485,37486,37514,37515,37543],{},"In this determinant, the first ",[17,37487,37489,37502],{"className":37488,"translate":21},[20],[17,37490,37492],{"className":37491},[25],[27,37493,37494],{"xmlns":29},[31,37495,37496,37500],{},[34,37497,37498],{},[37,37499,80],{},[41,37501,80],{"encoding":43},[17,37503,37505],{"className":37504,"ariaHidden":48},[47],[17,37506,37508,37511],{"className":37507},[52],[17,37509],{"className":37510,"style":92},[56],[17,37512,80],{"className":37513},[61,62]," rows contain only one non-zero ",[17,37516,37518,37531],{"className":37517,"translate":21},[20],[17,37519,37521],{"className":37520},[25],[27,37522,37523],{"xmlns":29},[31,37524,37525,37529],{},[34,37526,37527],{},[37,37528,80],{},[41,37530,80],{"encoding":43},[17,37532,37534],{"className":37533,"ariaHidden":48},[47],[17,37535,37537,37540],{"className":37536},[52],[17,37538],{"className":37539,"style":92},[56],[17,37541,80],{"className":37542},[61,62],"-order minor. Therefore, by Laplace's theorem,",[17,37545,37547],{"className":37546,"translate":21},[807],[17,37548,37550,37981],{"className":37549,"translate":21},[20],[17,37551,37553],{"className":37552},[25],[27,37554,37555],{"xmlns":29,"display":816},[31,37556,37557,37978],{},[34,37558,37559,37561,37563,37749,37751,37753,37755,37757,37760,37762,37764,37766,37820,37822,37824,37826,37828,37830,37972,37974,37976],{},[37,37560,111],{},[204,37562,825],{},[34,37564,37565,37567,37747],{},[204,37566,830],{"fence":48},[832,37568,37569,37611,37653,37697],{"rowspacing":834,"columnalign":19968,"columnspacing":836},[838,37570,37571,37581,37591,37597],{},[841,37572,37573],{},[844,37574,37575],{"scriptlevel":846,"displaystyle":621},[848,37576,37577,37579],{},[37,37578,25165],{},[408,37580,855],{},[841,37582,37583],{},[844,37584,37585],{"scriptlevel":846,"displaystyle":621},[848,37586,37587,37589],{},[37,37588,25165],{},[408,37590,866],{},[841,37592,37593],{},[844,37594,37595],{"scriptlevel":846,"displaystyle":621},[204,37596,3905],{"lspace":694,"rspace":694},[841,37598,37599],{},[844,37600,37601],{"scriptlevel":846,"displaystyle":621},[848,37602,37603,37605],{},[37,37604,25165],{},[34,37606,37607,37609],{},[408,37608,3544],{},[37,37610,80],{},[838,37612,37613,37623,37633,37639],{},[841,37614,37615],{},[844,37616,37617],{"scriptlevel":846,"displaystyle":621},[848,37618,37619,37621],{},[37,37620,25165],{},[408,37622,912],{},[841,37624,37625],{},[844,37626,37627],{"scriptlevel":846,"displaystyle":621},[848,37628,37629,37631],{},[37,37630,25165],{},[408,37632,923],{},[841,37634,37635],{},[844,37636,37637],{"scriptlevel":846,"displaystyle":621},[204,37638,3905],{"lspace":694,"rspace":694},[841,37640,37641],{},[844,37642,37643],{"scriptlevel":846,"displaystyle":621},[848,37644,37645,37647],{},[37,37646,25165],{},[34,37648,37649,37651],{},[408,37650,410],{},[37,37652,80],{},[838,37654,37655,37667,37679,37685],{},[841,37656,37657],{},[844,37658,37659],{"scriptlevel":846,"displaystyle":621},[34,37660,37661,37663],{},[37,37662,967],{"mathvariant":693},[969,37664,37665],{"height":694,"voffset":694},[228,37666],{"mathbackground":973,"width":694,"height":974},[841,37668,37669],{},[844,37670,37671],{"scriptlevel":846,"displaystyle":621},[34,37672,37673,37675],{},[37,37674,967],{"mathvariant":693},[969,37676,37677],{"height":694,"voffset":694},[228,37678],{"mathbackground":973,"width":694,"height":974},[841,37680,37681],{},[844,37682,37683],{"scriptlevel":846,"displaystyle":621},[34,37684],{},[841,37686,37687],{},[844,37688,37689],{"scriptlevel":846,"displaystyle":621},[34,37690,37691,37693],{},[37,37692,967],{"mathvariant":693},[969,37694,37695],{"height":694,"voffset":694},[228,37696],{"mathbackground":973,"width":694,"height":974},[838,37698,37699,37713,37727,37733],{},[841,37700,37701],{},[844,37702,37703],{"scriptlevel":846,"displaystyle":621},[848,37704,37705,37707],{},[37,37706,25165],{},[34,37708,37709,37711],{},[37,37710,80],{},[408,37712,3544],{},[841,37714,37715],{},[844,37716,37717],{"scriptlevel":846,"displaystyle":621},[848,37718,37719,37721],{},[37,37720,25165],{},[34,37722,37723,37725],{},[37,37724,80],{},[408,37726,410],{},[841,37728,37729],{},[844,37730,37731],{"scriptlevel":846,"displaystyle":621},[204,37732,3905],{"lspace":694,"rspace":694},[841,37734,37735],{},[844,37736,37737],{"scriptlevel":846,"displaystyle":621},[848,37738,37739,37741],{},[37,37740,25165],{},[34,37742,37743,37745],{},[37,37744,80],{},[37,37746,80],{},[204,37748,830],{"fence":48},[3907,37750,3923],{},[228,37752],{"linebreak":1087},[3907,37754,3923],{},[228,37756],{"linebreak":1087},[204,37758,37759],{},"∗",[204,37761,622],{"stretchy":621},[204,37763,627],{},[408,37765,3544],{},[403,37767,37768,37770],{},[204,37769,632],{"stretchy":621},[34,37771,37772,37774,37776,37778,37780,37782,37784,37786,37788,37790,37792,37794,37796,37798,37800,37802,37804,37806,37808,37810,37812,37814,37816,37818],{},[204,37773,622],{"stretchy":621},[408,37775,3544],{},[204,37777,3557],{},[408,37779,410],{},[204,37781,3557],{},[204,37783,3905],{},[204,37785,3557],{},[37,37787,80],{},[204,37789,632],{"stretchy":621},[204,37791,3557],{},[204,37793,622],{"stretchy":621},[37,37795,80],{},[204,37797,3557],{},[408,37799,3544],{},[204,37801,3557],{},[37,37803,80],{},[204,37805,3557],{},[408,37807,410],{},[204,37809,3557],{},[204,37811,3905],{},[204,37813,3557],{},[408,37815,410],{},[37,37817,80],{},[204,37819,632],{"stretchy":621},[3907,37821,3923],{},[228,37823],{"linebreak":1087},[3907,37825,3923],{},[228,37827],{"linebreak":1087},[204,37829,37759],{},[34,37831,37832,37834,37970],{},[204,37833,830],{"fence":48},[832,37835,37836,37866,37896,37940],{"rowspacing":834,"columnalign":19968,"columnspacing":836},[838,37837,37838,37848,37854,37860],{},[841,37839,37840],{},[844,37841,37842],{"scriptlevel":846,"displaystyle":621},[34,37843,37844,37846],{},[204,37845,627],{},[408,37847,3544],{},[841,37849,37850],{},[844,37851,37852],{"scriptlevel":846,"displaystyle":621},[408,37853,846],{},[841,37855,37856],{},[844,37857,37858],{"scriptlevel":846,"displaystyle":621},[204,37859,3905],{"lspace":694,"rspace":694},[841,37861,37862],{},[844,37863,37864],{"scriptlevel":846,"displaystyle":621},[408,37865,846],{},[838,37867,37868,37874,37884,37890],{},[841,37869,37870],{},[844,37871,37872],{"scriptlevel":846,"displaystyle":621},[408,37873,846],{},[841,37875,37876],{},[844,37877,37878],{"scriptlevel":846,"displaystyle":621},[34,37879,37880,37882],{},[204,37881,627],{},[408,37883,3544],{},[841,37885,37886],{},[844,37887,37888],{"scriptlevel":846,"displaystyle":621},[204,37889,3905],{"lspace":694,"rspace":694},[841,37891,37892],{},[844,37893,37894],{"scriptlevel":846,"displaystyle":621},[408,37895,846],{},[838,37897,37898,37910,37922,37928],{},[841,37899,37900],{},[844,37901,37902],{"scriptlevel":846,"displaystyle":621},[34,37903,37904,37906],{},[37,37905,967],{"mathvariant":693},[969,37907,37908],{"height":694,"voffset":694},[228,37909],{"mathbackground":973,"width":694,"height":974},[841,37911,37912],{},[844,37913,37914],{"scriptlevel":846,"displaystyle":621},[34,37915,37916,37918],{},[37,37917,967],{"mathvariant":693},[969,37919,37920],{"height":694,"voffset":694},[228,37921],{"mathbackground":973,"width":694,"height":974},[841,37923,37924],{},[844,37925,37926],{"scriptlevel":846,"displaystyle":621},[34,37927],{},[841,37929,37930],{},[844,37931,37932],{"scriptlevel":846,"displaystyle":621},[34,37933,37934,37936],{},[37,37935,967],{"mathvariant":693},[969,37937,37938],{"height":694,"voffset":694},[228,37939],{"mathbackground":973,"width":694,"height":974},[838,37941,37942,37948,37954,37960],{},[841,37943,37944],{},[844,37945,37946],{"scriptlevel":846,"displaystyle":621},[408,37947,846],{},[841,37949,37950],{},[844,37951,37952],{"scriptlevel":846,"displaystyle":621},[408,37953,846],{},[841,37955,37956],{},[844,37957,37958],{"scriptlevel":846,"displaystyle":621},[204,37959,3905],{"lspace":694,"rspace":694},[841,37961,37962],{},[844,37963,37964],{"scriptlevel":846,"displaystyle":621},[34,37965,37966,37968],{},[204,37967,627],{},[408,37969,3544],{},[204,37971,830],{"fence":48},[204,37973,825],{},[37,37975,18797],{},[37,37977,798],{"mathvariant":693},[41,37979,37980],{"encoding":43},"D = \\begin{vmatrix}  c_{11} &c_{12} & \\cdots &c_{1n} \\\\ c_{21}  &c_{22}  &\\cdots  & c_{2n}\\\\ \\vdots  & \\vdots &  &\\vdots \\\\  c_{n1} & c_{n2} &\\cdots  & c_{nn}\\end{vmatrix} ~\\\\ ~\\\\ \\ast (-1)^{(1+2+\\cdots+n)+(n+1+n+2+\\cdots +2n)} ~\\\\ ~\\\\ \\ast \\begin{vmatrix} -1  & 0 & \\cdots & 0\\\\  0 & -1 & \\cdots & 0\\\\ \\vdots  & \\vdots &  &\\vdots \\\\   0& 0 & \\cdots &-1 \\end{vmatrix} = 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(kkk-th Order Minor): In an nnn-th order determinant DDD, a kkk-th order minor MMM is a k×kk \\times kk×k determinant formed by selecting any kkk rows and kkk columns from DDD (k≤nk \\le nk≤n), and taking the k2k^2k2 elements at the intersections of these rows and columns in their original order.",false,true,"md",null,{"author":39356},"Ethan Zhang","2024-08-27","\u002Fblog\u002F2024\u002F2024-08-27-the-laplace-theorem-for-determinant",{"title":5,"description":39350},"blog\u002F2024\u002F2024-08-27-the-laplace-theorem-for-determinant","Laplace's theorem in determinants and its applications",[39363],"mathematics","tZnYlGApswjC70LqF9GPRiwNWCDVWKZfUwDeQyjHcEU",1777090297653]