Question

利用性质计算下列行列式 X-2 X-1 X-2 X-3 2X-2 2X-1 2X-2 2X-3 3X-3 3X-2 4X-5 3X-5 4X 4X-3 5X-7 4X-3

Answer 1

x-2 x-1 x-2 x-3
2x-2 2x-1 2x-2 2x-3
3x-3 3x-2 4x-5 3x-5
4x 4x-3 5x-7 4x-3

c1-c4,c2-c4,c3-c4
1 2 1 x-3
1 2 1 2x-3
2 3 x 3x-5
3 0 x-4 4x-3

r2-r1, r3-2r1, r4-3r1
1 2 1 x-3
0 0 0 x
0 -1 x-2 x+1
0 -6 x-7 x+3

r4-6r3
1 2 1 x-3
0 0 0 x
0 -1 x-2 x+1
0 0 -5x+5 -5x-3

r2<->r3, r3<->r4
1 2 1 x-3
0 -1 x-2 x+1
0 0 -5x+5 -5x-3
0 0 0 x

行列式 = x(5x-5).