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

解：
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
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