Multiplying a matrix by the other one is zero. Are those two matrice in null space of each other?

Question

We have two matrices $A$ and $B$, and we know that $AB=0$. Does it mean that $A$ is the null space of $B$?

Answer 1

It implies that each column of $B$ is in the null space of $A$.

Since the null space is a set of vectors and $B$ is a matrix, $B$ itself cannot be in the null space. It is not guaranteed either that the columns of $B$ will span the null space, although that is possible.

Answer 2

Assuming that they are both $n\times n$ matrices over a field $F$, the null space of $B$ is a vector subspace of $F^n$, whereas $A$ is not such a subspace. So, the answer is negative.