If $A$ and $B$ are nonzero matrices, their product $AB$ can still be the zero matrix.

True

Matrices have zero divisors: e.g. $\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix} = 0$. Rows of $A$ can be orthogonal to columns of $B$ everywhere, a possibility plain numbers never allowed.