# Inverse Matrices

• February 26th 2009, 11:55 AM
needshelp12
Inverse Matrices
Hello Everyone,

Really confused by this question, it might just be the wording of the question as a whole, but I was hoping someone would be able to give me some insight or clarification?

If
A, B and C are n x n invertible matrices, is there a matrix X that satisfies the matrix equation AC^-1(2X + A)AB^-1 = In? If so, find X.

Thanks

• February 26th 2009, 01:30 PM
HallsofIvy
Quote:

Originally Posted by needshelp12
Hello Everyone,

Really confused by this question, it might just be the wording of the question as a whole, but I was hoping someone would be able to give me some insight or clarification?

If
A, B and C are n x n invertible matrices, is there a matrix X that satisfies the matrix equation AC^-1(2X + A)AB^-1 = In? If so, find X.

Thanks

Solve for X just like you would any linear equation. The only thing to be careful about is that multiplication of matrices is not commutative so you have to be careful to multiply on the correct side. To get you started, looking at $AC^{-1}(2X+ A)AB^{-1}= I_n$ and seeing that "A" on the left, I would multiply on the left by $A^{-1}$ to get $C^{-1}(2X+ A)AB^{-1}= A^{-1}I_n= A^{-1}$. Seeing the " $B^{-1}$" on the right I would multiply on the right by B: $C^{-1}(2X+ A)A= A^{-1}B$. Continue!