Linear algebra proof regarding matrices

Question

I'd like a hint rather than a full solution.

The problem I am considering is the following:

$X$ is an $n\times m$ matrix $Y$ is $m\times n$

Show that

$(I - XY)^{-1}\cdot X = X\cdot(I - YX)^{-1}$

The first $I$ is of dimension $n$ and the second $I$ is of dimension $m$.

Answer 1

Hint

Verify that $$(I-XY)X=X(I-YX)$$