I used this http://en.wikipedia.org/wiki/Matrix_inverse#Blockwise_inversion formula to get the inverse of a partitioned matrix, and it works great. What I don't understand is why exactly it works.
If I have a matrix
$\left(\begin{array}{cc}A& B\\C& D\end{array}\right)$
and its inverse $\left(\begin{array}{cc}W& X\\Y& Z\end{array}\right)$
I can see that AW + BY = I, CX + DZ = I, and the other products are zero matrices, but when I try to use these relationships to build the blockwise formula back up, I don't get it right. Can anyone prove this formula to me?