求n阶行列式的值第一行(a,0,...,0,1)第二行(0,a,...,0,0)第n-1行(0,0,...,a,0)第n行(1,0,...,0,a)

Question

求n阶行列式的值第一行(a,0,...,0,1)第二行(0,a,...,0,0)第n-1行(0,0,...,a,0)第n行(1,0,...,0,a)

Answer 1

按照第1列展开：
a,0,...,0,1 a,0,...,0 0,...,0,1
0,a,...,0,0 0,a,...,0 a,...,0,0
.=a* .+(-1)^(n+1)* .
0,0,...,a,0 0,...,a,0 0,.a,0,0
1,0,...,0,a 0,...,0,a 0,...,a,0
=a^n-a^(n-2) (后面的n-1阶行列式最后一列起,与前列交换,共n-2次,乘以(-1)^(n-2）