Trying to prove that if A > 0, then $M$ is positive semidefinite if and only if $C=B^*$ and $D-CA^{-1}B \ge 0$

I have written
 M=\begin{pmatrix} A&B\\B^*&D \end{pmatrix}= \begin{pmatrix} 1&BD^{-1}\\0&1 \end{pmatrix}\begin{pmatrix} A-BD^{-1}B^*&0\\0&D \end{pmatrix}\begin{pmatrix} 1&BD^{-1}\\0&1 \end{pmatrix}^* = QPQ^*

but this isn't quite right...I think the last matrix needs to be

 \begin{pmatrix} 1&0\\B^*D^{-1}&1 \end{pmatrix}

but then I can't write it as  QPQ^*, can I? Or are they the same?