求行列式的值x+123...n1x+23...n12x+3...n.........123...x+n谢谢给出解答过程

Question

求行列式的值
x+1 2 3 ... n
1 x+2 3 ... n
1 2 x+3 ... n
.........
1 2 3 ... x+n
谢谢给出解答过程

Answer 1

x+1 2 3 ...n
1 x+2 3 ...n
1 2 x+3 ...n
.
1 2 3 ...x+n
x+n(n+1)/2 2 3.n 1 2 3.n
x+n(n+1)/2 x+2 3.n 1 x+2 3.n
= x+n(n+1)/2 2 x+3.n =[x+n(n+1)/2] 1 2 x+3.n
..
x+n(n+1)/2 2 3.x+n 1 2 3.x+n
1 2 3.n
0 x 0.0
=[x+n(n+1)/2] 0 0 x.0
.
0 0 0.x
=[x+n(n+1)/2]*x^(n-1)
=x^n+n(n+1)x^(n+1)/2