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?

Since $AB-I$ is invertible, there exists an $X$ such that $(AB-I)X=I$. 
Note that $(A-B^{-1})B=AB-I$. Hence $(A-B^{-1})BX=I$.
