Inverse of a $2 \times 2$ block matrix with a zero diagonal block

Question

I want to compute the inverse of the $2\times2$ block matrix

$$\left(\begin{array}{cc} A & P\\ P^T & 0\end{array}\right),$$

where $A$ is an $n\times n$ matrix and $P$ is an $n \times m$ matrix.

I only come across formulas for $$ \left(\begin{array}{cc} A & B\\ C & D\\ \end{array}\right)^{-1} $$ that involve $D^{-1}$. However, in my case $D=0$, such that $D$ has no inverse and I don't know how to continue.

Answer 1

One of the formulas here does not involve $D^{-1}$