I was wondering how to expand the inverse of the difference of two matrices multiplied by another matrix, i.e. a matrix of the form:
$C\left ( A-B \right )^{-1}$
Is there a certain method applicable to any case as the one above?
Also would the following case follow the same principle?
$C\left ( A-B \right )^{-1}D$
You can use the Woodbury matrix identity if you want to expand $(A-B)^{-1}$. According to this identity, if $A$ and $B$ are invertible, it holds that $$(A-B)^{-1}=A^{-1}-A^{-1}(-B^{-1}+A^{-1})A^{-1}.$$ You can then left-multiply by $C$ to get the expansion. In the special case that $B$ is a rank-one matrix, then you can use the Sherman-Morrison formula to expand the inverse.