I have searched for the above topic and found some results, but the answer I am looking for is not found anywhere. Here is my question:

> 
> Given $A\_{m \times n}$ matrix with rank $m$, and $B\_{n \times p}$ matrix with rank $p$, where $n > p \geq m$. I know that 
>  $$
>  \operatorname{rank}(AB) \leq 
>  \min\left(\operatorname{rank}(A),\operatorname{rank}(B)\right)
> $$
>  What I want to know is if this expression holds for equality. I.e is this expression
>  $$
>  \operatorname{rank}(AB) = 
>  \min\left(\operatorname{rank}(A),\operatorname{rank}(B)\right) = m
> $$
>  correct? If yes, how can it be proved?
> 
> 
>