Given a matrix $X$, let $D$ be a diagonal matrix whose diagonal elements are row sums of $X$, let $I$ be an identity matrix. Now I have a resultant matrix of $Y=(I-X)^{-1}$, and I would like to compute $Z=(I-X-DX)^{-1}$.
Is there some methods to re-use the result of $Y$ to get the result of $Z$ ? In other words, is it possible to express $Z$ in terms of $Y$ ?
So $Y^{-1}=I-X$
$X=I-Y^{-1}$
$Z=((I-X)-DX)^{-1}$
$Z=(Y^{-1}-D(I-Y^{-1}))^{-1}$