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 ${\rm{rank}}(P) = {\rm{rank}}(A)+{\rm{rank}}(D)$?
Best Answer
No, not necessarily.
For example if $m=n$ with $B=D$ and $C=A$, we get $\text{rank}(P)=n$,$\;$not $2n$.