Thread: invertible matrices

Please help!

Suppose A,B,& X are n by n matrices, with A, X, & A-AX invertable, & suppose:

(A-AX)^-1=X^-1*B

I know B is invertable

But I can't solve for X.

Any Help

Suppose (A-AX)^-1=X^-1*B.
Multiply on the left by X:
X(A-AX)^-1 = B.
Multiply on the right by (A-AX):
X = B(A-AX).
X + BAX = BA
X(I+BA) = BA.
(X-I)(I+BA) = BA - (I+BA) = -I.
Assume I+BA is invertible.
X-I = -(I+BA)^-1
X = I - (I+BA)^-1.
If I+BA is not invertible there's no solution in X.