行列式主对角线都是0其余去下

Question

主对角线上都是0第一行第一列除了第一个数外都是1其余均为x的行列式怎么求解

Answer 1

Dn=|0 1 1 1 ... 1|
1 0 x x ... x
1 x 0 x ... x
1 x x 0 ... x
..............
1 x x x ... 0

r2-r1*x、r3-r1*x、...、r n-r1*x

=|0 1 1 1 ... 1|
1 -x 0 0 ... 0
1 0 -x 0 ... 0
1 0 0 -x ... 0
...................
1 0 0 0 ... -x

r1+r2/x+r3/x+...+r n/x

=|0+1/x+1/x+...+1/x 0 0 0 ... 0|
1 -x 0 0 ... 0
1 0 -x 0 ... 0
1 0 0 -x ... 0
......................
1 0 0 0 ... -x

=[(n-1)/x](-x)^(n-1)
=[(-1)^(n-1)]*(n-1)*[x^(n-2)]