TITLE: Prove that if $\operatorname{rank}A=n$, then $\operatorname{rank}AB=\operatorname{rank}B$
QUESTION [10 upvotes]: Let $A \in M_{m\times n}(\mathbb{R})$ and $B \in M_{n\times p}(\mathbb{R})$.
Prove that if $\operatorname{rank}(A)=n$ then $\operatorname{rank}(AB)=\operatorname{rank}(B)$.

I tried to start with definitions finding that $n \le m$, but didn't know what to do with $AB$.
Please help, thank you!

REPLY [3 votes]: Hint: $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ is easy.
If $A$ is $m \times n$ and $\operatorname{rank}(A)=n$, then $A$ has a left inverse.
Call this inverse $C$ and use
$$\operatorname{rank}(CAB) \leq \operatorname{rank}(AB).$$