If the product of two $n \times n$ matrices $A$ and $B$ is zero ie: $AB = 0$
Then either $\det(A)$ or $\det(B)$ must be zero.
What additional conditions on $A$ and $B$ would be sufficient ? Clearly the condition that either determinant has to be zero is not sufficient, as there could be some $A$ or $B$ whose determinant is zero but $AB \not= 0$
Best Answer
$AB=0$ iff the columnspace of $B$ is contained in the nullspace of $A$ (or if you are instead viewing them as acting on row vectors, the rowspace of $A$ contained in the kernel of $B$).
The determinant of one of them being zero is just a very weak indication that at least one of them has a nonzero nullspace. Dimensions are not going to tell you very much: one needs to look at more detail at the interrelationships of these subspaces, rather than just their dimensions.