Let $A$ be a $9 \times 4$ matrix and $B$ be a $7 \times 3$ matrix. Is there a matrix $X \ne 0$ such that $AXB=0$?
I'm not sure how to solve this. If $A$ or $B^T$ have a non-trivial null-space, it makes things easy, but if they don't, how would I proceed?
Hint 1
What if $A$ really has one row and $B$ only has one column? then you are asking given vectors $\vec{a} \in \mathbb{R}^n$ and $\vec{b} \in \mathbb{R}^m$, is there a matrix $X \in \mathbb{R}^{n \times m}$ such that $\vec{a}^T X \vec{b} = 0$. What are your thoughts about this smaller problem?
Hint 2
Given some vector $\vec{b} \in \mathbb{R}^m$, can you construct another vector $x \in \mathbb{R}^m$ such that $\vec{x} \cdot \vec{b} = 0$? In fact the set of such vectors defines a vector space, commonly denoted $b^\perp$.