I'm struggling with the proof of the following proposition:
Given a $n\times n$ symmetric, positive-semidefinite matrix $\Omega$, a $n\times k$ matrix $X$ such that $rank(X)=k$, and an invertible matrix $Q$ such that: $$\Omega X=XQ$$
Prove the following: ${{(X'{{\Omega }^{-1}}X)}^{-1}}X'{{\Omega }^{-1}}={{(X'X)}^{-1}}X'$
Here's my work before getting stuck:
I also tried playing with the Cholesky factorization of $\Omega$ but always ended up getting stuck with an irreducible expression. What am I missing? (for context, this problem is from a graduate-level econometrics course)