Matrix Algebra: Prove this holds true

[Hendrik]

I have a problem with one of my homework questions. First of all, I'm not even sure what is being asked since a hint in my textbook and the question itself are referring to different statements. The actual question says:

Let \(\displaystyle \Omega>0\) (i.e. pd) and let \(\displaystyle X: n\)x\(\displaystyle k\) have rank \(\displaystyle k\) (i.e. full column rank). Show:

\(\displaystyle (X'\Omega^{-1}X)^{-1}\leq(X'X)^{-1}X'\Omega X(X'X)^{-1}\)

However, my textbook refers to:

\(\displaystyle X'\Omega^{-1}X\leq(X'X)^{-1}X'\Omega X(X'X)^{-1}\)

These cannot both hold true, can they? Anyway, neither of them I'm able to prove.

The hint states:
You might want to use the property that if \(\displaystyle P\) is a projection matrix, then \(\displaystyle 0\leq P\leq I\). So, for example, \(\displaystyle A'PA\leq A'A\).

I get this, but don't know how to use it. Starting with \(\displaystyle P\leq I\), (where \(\displaystyle P\) is \(\displaystyle P_X=X(X'X)^{-1}X'\) or \(\displaystyle P_{\Omega}=\Omega (\Omega' \Omega)^{-1} \Omega'\) I tried applying several manipulations but I didn't manage to end up with one of the above presented statements.

I hope someone can help me!