TITLE: Prove that the rank of the block matrix is rank $A$ + rank $B$
QUESTION [9 upvotes]: Assume $A,B$ are $n \times n$ matrices.
Prove that the rank of the block matrix that is defined as the following
$$
\pmatrix{A& AB\\B&B+B^2}
$$
 is equal to $ \mbox{rank} A +\mbox{rank} B$.
I have absolutely no clue...

REPLY [8 votes]: Row/column reduction of block matrices works here. First substract $B$ times the first column from the second column. Then subtract the second column from the first, giving you the matrix 
$$C = \pmatrix{A& 0\\ 0 & B}$$
This is where the matrices in the answer by Humanity come from. Note that the row operations will not change the rank of the matrix. Hence the matrix has same rank as $C$, and it is not too difficult to see that $C$ has rank equal to $\operatorname{rank} A + \operatorname{rank} B$.