Let $A$ be a given fixed $n \times m$ matrix. We also consider matrices $B$ of dimension $m \times p$. I am interested in those matrices $A$, for which for all $B \in \mathbb R^{m \times p}$ the product $AB$ has at least one row $r_i$, such that there is not a single 0 in the row $r_i$.

Is there already terminology for this? I would like to have a useful characterization of $A$ based on linear algebra. Thank you.