解法1.用初等行变换将(A,E)化为(E,A^-1)
(A,E) =
1 2 1 -2 1 0 0 0
0 5 3 -2 0 1 0 0
0 0 3 5 0 0 1 0
0 0 0 3 0 0 0 1
r4*(1/3),r1+2r4,r2+2r4,r3-5r4
1 2 1 0 1 0 0 2/3
0 5 3 0 0 1 0 2/3
0 0 3 0 0 0 1 -5/3
0 0 0 1 0 0 0 1/3
r2-r3,r3*(1/3),r1-r3
1 2 0 0 1 0 -1/3 11/9
0 5 0 0 0 1 -1 7/3
0 0 1 0 0 0 1/3 -5/9
0 0 0 1 0 0 0 1/3
r2*(1/5),r1-2r2
1 0 0 0 1 -2/5 1/15 13/45
0 1 0 0 0 1/5 -1/5 7/15
0 0 1 0 0 0 1/3 -5/9
0 0 0 1 0 0 0 1/3
A^-1 =
-2/5 1/15 13/45
1/5 -1/5 7/15
0 1/3 -5/9
0 0 1/3
解2.用分块矩阵方法求逆
A =
B C
0 D
当B,D可逆时A也可逆,且 A^-1 =
B^-1 -B^-1CD^-1
0 D^-1