Find the inverse of a matrix Assume $A,B,AB-I$ are invertible.   What is the inverse of $A-B^{-1}$ ?    I guessed some matrices but they don't work.  Is there a systematic way to figure this out without guessing?

Let $J$ be the inverse of $AB-I$, so we have $ (AB-I)J=J(AB-I)=I$. After expanding this, we get $ABJ-J=JAB-J=I$. Now
$$
(A-B^{-1})(B J)=ABJ-J=I
$$
so $BJ$ is the right inverse. It's up to you to find the left inverse.
