Rank of concatenation of two full row rank matrices

Question

Let

$$ C_{a\times (b+c)} := \begin{bmatrix} A_{a\times b} & B_{a\times c} \end{bmatrix} $$

where matrices $A$ and $B$ are full row rank, i.e., $\mbox{rank}(A) = \mbox{rank} (B) = a$. What is the rank of $C$?

I assume $\mbox{rank} (C)=a$, but I cannot find a proof for this.

Answer 1

Can the rank of $C$ be less than $a$, given that both $A$ and $B$ have rank $a$? Can the rank of a matrix be greater than the number of its rows? Answering these questions should help you answer yours.