若矩阵B满足A+B=AB。求B

Question

1 -1 1
A= -1 1 -1
1 -1 1

Answer 1

A+B=AB
移项合并得到A=(A-E)B
然后两边左乘一个（A-E）的逆就可以了
所以这道题其实就是求逆

Answer 2

解: 由已知, (A-E)B=A.
(A-E,A)=
0 -1 1 1 -1 1
-1 0 -1 -1 1 -1
1 -1 0 1 -1 1

r2+r3
0 -1 1 1 -1 1
0 -1 -1 0 0 0
1 -1 0 1 -1 1

r1-r2, r3+r2, r2*(-1)
0 0 2 1 -1 1
0 1 1 0 0 0
1 0 1 1 -1 1

r1*(1/2),r2-r1,r3-r1
0 0 1 1/2 -1/2 1/2
0 1 0 -1/2 1/2 -1/2
1 0 0 1/2 -1/2 1/2

r1<->r3
1 0 0 1/2 -1/2 1/2
0 1 0 -1/2 1/2 -1/2
0 0 1 1/2 -1/2 1/2

所以 B=
1/2 -1/2 1/2
-1/2 1/2 -1/2
1/2 -1/2 1/2

Answer 3

A+B=AB
A=AB-B=(A-E)B
B=(A-E)逆*A
(A-E)逆=1/2[0 -1 1;-1 -1 -1;1 -1-1]
B=[1,-1,1;-1/2 1/2 -1/2; 1/2 -1/2 -1/2]本回答被网友采纳