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

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- July 21st 2005, 07:23 AMjackbell123invertible 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 - July 21st 2005, 01:32 PMrgep
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.