$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____?

Question

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____?

This is a fill-in-the-blank problem that I found in my paper, but I don't have this answer.

Answer 1

Perhaps the intended answer is "square matrices", since $\text{det}(A)$ is undefined when $A$ is not square.

Answer 2

It is clear that $A$ and $B$ must be square matrices of the same dimension. However, since you do not specify on which set they are defined I would comment as follow.

Let $R$ be a ring. Then we define $M_{n\times n}(R)$ to be the set of $n$-by-$n$ matrices over $R$. It is possible to define the determinant of $A\in M_{n\times n}(R)$ as usual (the sum over the permutation in $S_n$..). Then we have that $\mathrm{det}(AB)=\mathrm{det}(A)\mathrm{det}(B)$ if $R$ is commutative ring. The equality is in general false if we drop the commutativity hypothesis on $R$.