If the product of two matrices is a null matrix, when can we say one of them is a null matrix?

Question

$A$ and $B$ are two rectangular matrices of size $m \times n$ and $n \times p$, respectively (where $n > m,p$). Also $A$ is full rank. If $AB = 0$ can we say $B = 0$?

Answer 1

The simplest example is to pick $ A $ to have one row and $ B $ one column ($ m = p = 1 $). $ A B $ then becomes a dot product, and we know that for any two orthogonal vectors their dot product is $ 0 $.

This is the kind of example that is important to think about, since it shows that a lot of what you know about algebra with scalars does not carry over to algebra with matrices.