rank of block matrix whose diagonal blocks are invertible

Question

Suppose I have a block matrix $$P = \begin{bmatrix} A & B \\ C & D\end{bmatrix},$$ where $A\in\mathbb{R}^{n\times n}$ and $D\in\mathbb{R}^{m\times m}$ are invertible. $B\in\mathbb{R}^{n\times m}$ and $C\in\mathbb{R}^{m\times n}$. Then what is the rank of the block matrix $P$?

Do I have ${\rm{rank}}(P) = {\rm{rank}}(A)+{\rm{rank}}(D)$?

Answer 1

No, not necessarily.

For example if $m=n$ with $B=D$ and $C=A$, we get $\text{rank}(P)=n$,$\;$not $2n$.

Answer 2

No. This fails already when $n=m=1$. For instance $$P=\begin{bmatrix}1&1\\1&1\end{bmatrix}$$ has rank $1$. The only guarantee you have is that $$\operatorname{rank}P\geq\max\{\operatorname{rank}A,\operatorname{rank}D\}.$$