How to prove the statement about the rank of a block matrix?

Question

Let $A$ and $B$ be real matrices with the same number of rows. Prove that:

$$\mbox{rank} \begin{bmatrix} A & B\\ 2A & -5B\end{bmatrix} = \mbox{rank}(A) + \mbox{rank}(B)$$

---

I have no idea how to approach the problem. Could you give me a hint?

Answer 1

Subtract the double of the first block row from the second one, we get $$ \pmatrix{A&B\\ 0&-7B}. $$ Then divide the second block row by $-7$ to obtain $$ \pmatrix{A&B\\ 0&B}. $$ Finally, subtract the second block row from the first block row to obtain $$ M=\pmatrix{A&0\\ 0&B}. $$ If you can prove that $\operatorname{rank}(M)=\operatorname{rank}(A)+\operatorname{rank}(B)$, then you are done, because the rank of a matrix is unaffected by elementary row operations.