1 2 2 2 2 3 1 2 1 1 1-1 1 0-2-6 用初等变换法求矩阵逆,

(A,E)=
1 2 2 2 1 0 0 0
2 3 1 2 0 1 0 0
1 1 1 -1 0 0 1 0
1 0 -2 -6 0 0 0 1
r1-r3,r2-2r3,r4-r3
0 1 1 3 1 0 -1 0
0 1 -1 4 0 1 -2 0
1 1 1 -1 0 0 1 0
0 -1 -3 -5 0 0 -1 1
r2-r1,r3-r1,r4+r1
0 1 1 3 1 0 -1 0
0 0 -2 1 -1 1 -1 0
1 0 0 -4 -1 0 2 0
0 0 -2 -2 1 0 -2 1
r2-r4,r4*(-1/2),r1-r4
0 1 0 2 3/2 0 -2 1/2
0 0 0 3 -2 1 1 -1
1 0 0 -4 -1 0 2 0
0 0 1 1 -1/2 0 1 -1/2
r2*(1/3),r1-2r2,r3+4r2,r4-r2
0 1 0 0 17/6 -2/3 -8/3 7/6
0 0 0 1 -2/3 1/3 1/3 -1/3
1 0 0 0 -11/3 4/3 10/3 -4/3
0 0 1 0 1/6 -1/3 2/3 -1/6
交换行
1 0 0 0 -11/3 4/3 10/3 -4/3
0 1 0 0 17/6 -2/3 -8/3 7/6
0 0 1 0 1/6 -1/3 2/3 -1/6
0 0 0 1 -2/3 1/3 1/3 -1/3
A^-1 =
-11/3 4/3 10/3 -4/3
17/6 -2/3 -8/3 7/6
1/6 -1/3 2/3 -1/6
-2/3 1/3 1/3 -1/3

使用左乘初等变换矩阵，使得其变成上三角阵；然后使用右乘初等变换矩阵，使其变成对角阵
然后对两边矩阵求逆即可
初等变换矩阵和对角阵的逆都是很好求的。