Question

计算一个n阶行列式，主对角线为a1到an，其余元素都为x

X≠ai i=1，2，......，n

Answer 1

|a1 x x ... x|
x a2 x ... x
x x a3 ... x
.................
x x x ... an

r2-r1、r3-r1、...、rn-r1
=|a1 x x .... x|
x-a1 a2-x 0 .... 0
x-a1 0 a3-x .... 0
............................
x-a1 0 0 .... an-x

r1-r2x/(a2-x)-r3x/(a3-x)-...-rnx(an-x)
=|m 0 0 ....... 0| 【m=a1+(a1-x)/(a2-x)+...+(a1-x)/(an-x)
x-a1 a2-x 0 ....... 0
x-a1 0 a3-x ....... 0
..............................
x-a1 0 0 ....... an-x 【是个《下三角》

=m*∏(ak-x) [k=2 to n]
=[a1+(a1-x)∑1/(ak-x)]*∏(ak-x) 【 k= 2 to n】