Property of matrix determinants

Question

So, recently my tutor showed through a couple of examples that if you take $2$ matrices say $A_{\;3 \times 2}$ and $B _{\;2 \times 3}$ then we will either have $\det. (AB) =0$ or $\det . (BA)=0$ and then challenged us to prove it. I have tried and even saw a couple of videos on linear algebra but I am nowhere closer to figuring it out. He said that the only thing that needs to be ensured was that both $AB$ and $BA$ exist. Any advice would be appreciated.

Answer 1

Hint The rank of a matrix product satisfies $$\operatorname{rank} (PQ) \leq \max \{\operatorname{rank} P, \operatorname{rank} Q\} .$$ If $A$ has size $m \times n$, $m \neq n$, and $B$ has size $n \times m$, what are the possible ranks of $A, B$, and what are the sizes of $AB, BA$?