用初等变换求下列矩阵的逆阵(1,1,-1;2,1,0;1,-1,0;)答案为(0,1/3,1/3;0,1/3,-2/3;-1,2/3,-1/3),

Question

用初等变换求下列矩阵的逆阵(1,1,-1;2,1,0;1,-1,0;)答案为(0,1/3,1/3;0,1/3,-2/3;-1,2/3,-1/3),

Answer 1

(A,E)=
1 1 -1 1 0 0
2 1 0 0 1 0
1 -1 0 0 0 1
r1-r2,r3+r2
-1 0 -1 1 -1 0
2 1 0 0 1 0
3 0 0 0 1 1
r3*(1/3),r1+r3,r2-2r3
0 0 -1 1 -2/3 1/3
0 1 0 0 1/3 -2/3
1 0 0 0 1/3 1/3
r1*(-1),r1r3
1 0 0 0 1/3 1/3
0 1 0 0 1/3 -2/3
0 0 1 -1 2/3 -1/3
所以 A^-1=
0 1/3 1/3
0 1/3 -2/3
-1 2/3 -1/3