The determinant behaves multiplicatively with respect to the usual matrix product $$ \det(AB) = \det(A)\det(B), $$ and also with respect to the Kronecker (or tensor) product of square matrices $$ \det(A\otimes B) = \det(A)^q \det(B)^p, $$ when $A$ and $B$ are $p\times p$ and $q \times q$ matrices, respectively.

Are there other natural types of matrix products under which the determinant behaves multiplicatively? To be completely precise, the property I need is that the determinant of the product is $0$ if and only if the determinant of at least one of its factors is $0$.