I was asked to prove the following in my homework:
$(I+PQ)^{-1}P = P(I+QP)^{-1}$
Is there any way to prove this equality without the complicatedness of the Woodbury formula? I understood that
$[(I+P)^{-1}P = P(I+P)^{-1}]$
but couldn't get my head around the PQ and QP part.
HINT
Multiply on the left with $I+PQ$ and on the right with $I + QP$
Assuming $P,Q\in\mathcal{M}_n$ and $P$ is invertible
$$\left(I+PQ\right)^{-1}P=\left(PIP^{-1}+PQPP^{-1}\right)^{-1}\,P=\left(P\left(IP^{-1}+QPP^{-1}\right)\right)^{-1}\,P=$$ $$=\left(IP^{-1}+QPP^{-1}\right)^{-1}P^{-1}P=\left(IP^{-1}+QPP^{-1}\right)^{-1}=$$ $$=\left(\left(I+QP\right)P^{-1}\right)^{-1}=P\left(I+QP\right)^{-1} $$