Question

求矩阵A= 0 2 1，1 -1 0，-1 2 1的逆矩阵

Answer 1

(A, E) =
[ 0 2 1 1 0 0]
[ 1 -1 0 0 1 0]
[-1 2 1 0 0 1]
初等行变换为
[ 1 -1 0 0 1 0]
[ 0 2 1 1 0 0]
[0 1 1 0 1 1]

初等行变换为
[ 1 0 1 0 2 1]
[0 1 1 0 1 1]
[ 0 0 -1 1 -2 -2]
初等行变换为
[ 1 0 0 1 0 -1]
[ 0 1 0 1 -1 -1]
[ 0 0 1 -1 2 2]
A^(-1) =
[ 1 0 -1]
[ 1 -1 -1]
[-1 2 2]

Answer 2

1/4 1/4 1/4
1/2 -1/2 1/2
-1/4 -1/4 3/4
方法：将原矩阵扩充为[A E],实行初等行变换成[E A^(-1)],即得逆矩阵.