${\rm rank}(BA)={\rm rank}(B)$ if $A \in \mathbb{R}^{n \times n}$ is invertible?

Question

I'm having some trouble with the following question:

Let $A, B \in \mathbb{R}^{n \times n}$ and let $A$ be invertible. Is it true that in this case $rank(BA)=rank(B)$?

I think that this statement is correct, but I'm unable to prove it.

My thoughts so far:

If $B$ is also invertible the statement clearly holds, since $GL_n(\mathbb{R})$ is a group.

For $B$ not invertible we immediately have the inequality $rank(BA) \leq rank(B)$ because the columns of $BA$ are linear combinations of the columns of $B$.

Now I've tried to prove the other inequality by contradiction, i.e. assuming that $rank(BA)<rank(B)$ and showing that this cannot be. But I can't complete this step.

Thanks in advance for any help!

Answer 1

Hint. If $A$ is invertible then the set of vectors $A{\bf x}$, where ${\bf x}\in{\Bbb R}^n$, is the whole of ${\Bbb R}^n$. So $${\rm im}(BA)=\{BA{\bf x}\mid {\bf x}\in{\Bbb R}^n\}=\{B{\bf y}\mid {\bf y}\in{\Bbb R}^n\}={\rm im}(B)\ .$$

This is more or less the whole answer, but see if you can provide the reason for each step.

Note: depending on the notation you are using in your course, ${\rm im}(B)$ is the same as ${\rm col}(B)$ or ${\rm CS}(B)$.

Answer 2

Hint: For any two matrices $A$ and $B$ the inequality $$ \DeclareMathOperator{rk}{rk}\rk(BA)\leq\rk(B) $$ holds since the columns of $BA$ are linear combinations of the columns of $B$. Since $\rk(X)=\rk(X^\top)$ we then also have $$ \rk(BA)=\rk((BA)^\top)=\rk(A^\top B^\top)\leq\rk(A^\top)=\rk(A) $$ Can you use this to finish your problem?