- #1

Cyborg31

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## Homework Statement

A, B, I-Y^-1, X, Y are n x n invertible matrices

(A(I-Y^-1))^-1 = YB

Solve for Y

## Homework Equations

## The Attempt at a Solution

(I - Y^-1)^-1A^-1 = YB

(I - Y^-1)^-1A^-1B^-1 = Y

A^-1B^-1 = (I - Y^-1)Y

A^-1B^-1 = Y - I

A^-1B^-1*(A^-1B^-1)^-1 = Y - I(A^-1B^-1)^-1

I = Y - (A^-1B^-1)^-1

Y^-1*I = Y^-1*Y - (A^-1B^-1)^-1

Y^-1 = I - (A^-1B^-1)^-1

Inverse both sides

Y = (I - (A^-1B^-1)^-1)^-1

or is it

Y = I^-1 - A^-1B^-1

Y = I - A^-1B^-1

Is this correct?