rank(AB) vs rank(BA)

Question

How should I evaluate this statement?
I am looking for a simple way to prove or disprove this statement within the context my linear algebra course.

If $A$ and $B$ are $4 \times 4$ matrices such that $\operatorname{rank}(AB) = 3$, then $\operatorname{rank}(BA) < 4$.

Answer 1

If the rank of $AB$ is not $4$, at least one matrix between $A$ and $B$ is not invertible, so the same happens for $BA$.

Answer 2

Hint: Look at the determinants.