Question

求解一道行列式的题目！！！

Answer 1

A(x)=
1+x 1 1 1
1 x 1 1
1 1 x 1
1 1 1 x

|A(x)|=
1 1 1 1
0 x 1 1
0 1 x 1
0 1 1 x
+
x 1 1 1
1 x 1 1
1 1 x 1
1 1 1 x
= (x+2)(x-1)^2 + (x+3)(x-1)^3
= (x^2+3x-1)(x-1)^2.

(A(-2),E)=
-1 1 1 1 1 0 0 0
1 -2 1 1 0 1 0 0
1 1 -2 1 0 0 1 0
1 1 1 -2 0 0 0 1
经初等行变换化为
1 0 0 0 0 1/3 1/3 1/3
0 1 0 0 1/3 -1/9 2/9 2/9
0 0 1 0 1/3 2/9 -1/9 2/9
0 0 0 1 1/3 2/9 2/9 -1/9
所以 A^-1(-2) =
0 1/3 1/3 1/3
1/3 -1/9 2/9 2/9
1/3 2/9 -1/9 2/9
1/3 2/9 2/9 -1/9