Expand the determinant according to the third column and calculate its value: the first row 1234, the second row 2341, the third row 3412, the fourth row 4123

Expand the determinant according to the third column and calculate its value: the first row 1234, the second row 2341, the third row 3412, the fourth row 4123

1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3
Expand by the third column;
2 3 1 1 2 4 1 2 4 1 2 3
3 3 4 2 - 4 3 4 2 + 2 3 1 - 2 2 3 1
4 1 3 4 1 3 4 1 3 3 4 2
①
2 3 1 0 0 1 -1 -2
3 4 2 = -1 -2 2 = =8-4=4
4 1 3 -2 -8 3 -2 -8
②
1 2 4 1 2 4 -2 -10
3 4 2 = 0 -2 -10 = =26-70=-54
4 1 3 0 -7 -13 -7 -13
③
1 2 4 1 2 4 -1 -7
2 3 1 = 0 -1 -7 = =13-49=-36
4 1 3 0 -7 -13 -7 -13
④
1 2 3 1 2 3 -1 -5
2 3 1 = 0 -1 -5 = =7-10=3
3 4 2 0 -2 -7 -2 -7
So determinant = 3 × 4-4 × (- 54) - 36-2 × 3 = 186

Determinant first row 0 a B C second row a 0 C B third row B C 0 a fourth row C B a 0

c1+c2+c3+c4a+b+c a b ca+b+c 0 c ba+b+c c 0 aa+b+c b a 0r2-r1,r3-r1,r4-r1a+b+c a b c0 -a c-b b-c0 c-a -b a-c0 b-a a-b -cc2+c3a+b+c a+b b c0 c-a-b c-b b-c0 c-a-b -b a-c0 0 a-b -cr3-r2a+b+c a+b b c0 c-a-...

Calculate | a | = | the first line a B C D | the second line - B a - D C | the third line - C D a - B | the fourth line - D - C B a |
Because the AAT (A and the transposmatrix are multiplied by the AAT (a) and the transposmatrix matrix, which is the multiplication of (a ^ 2 + B ^ 2 + C ^ 2 + C ^ 2 + C ^ 2 + C ^ 2 + C ^ 2 + D ^ 2) e, there is 124124124\124\124\124\124\124\\\\\\\\124\\\\\\\124\\124\\\\\124\ + D ^ 2) ^ 2 how does this come from? Where is a ^ 4? How to get the coefficient?

A ^ 4 is obtained by taking a from each line, and the reverse order number of this method is 0, so the sign is + and the coefficient is + 1, so | a | = (a ^ 2 + B ^ 2 + C ^ 2 + D ^ 2) ^ 2 instead of | a | = - (a ^ 2 + B ^ 2 + C ^ 2 + D ^ 2) ^ 2