Question

If A, B, C are invertible, then does there exist an X such that \[C^{-1}(A+X)B^{-1}=I\]?

Answer 1

What I did was isolate X, and found X = CB -A. I made sure this was satisfying the equation by replacing it in the original equation:

\[C^{-1}(A+X)B^{-1}=I\]\[C^{-1}(A+(CB-A))B^{-1}\]\[C^{-1}(A-A+CB)B^{-1}\]\[C^{-1}(CB)B^{-1}=II=I\]

Is this appropriate to say that Yes, X exists and is CB - A? I'm not sure why I'm not sure if this is the right way to do it.

Answer 2

It certainly seems like the right way to do it. As long as you checked that it works, you should be fine.

Answer 3

Oh ok thanks =]

Answer 4

you're welcome