For my question you need the definition of the projection matrix: $Proj = X(X'X)^{-1}X'$ and the rule $(AB)^{-1} = B^{-1}A^{-1}$
This leads me to believe the following:
$Proj = X(X'X)^{-1}X' = XX^{-1}X'^{-1}X' = I * I = I$
Surely I must be making a mistake somewhere since this would render the projection matrix useless. I can not see where what I'm missing though since its such a small problem. Can anyone point me to my mistake?
$$(AB)^{-1} = B^{-1}A^{-1}$$ works only if $A,B$ are both square invertible matrices