Suppose we have two fixed real matrices $A \in M_{n \times m}(\mathbb R)$ and $B \in M_{r \times n}(\mathbb R)$ with $n > m$ and $r < n$. Moreover, $A$ has full column rank $m$ and $B$ has full row rank $r$. I am interested in constructing a nontrivial matrix $X \in M_{m \times r}(\mathbb R)$ such that $AXB$ is upper/lower triangular with diagonal entries to be $0$.
I tried constructing some concrete examples for $A$ and $B$. It seems that for some $A$ and $B$, we can have $X$. For example, we can take $A =\pmatrix{0 & 0 \\ 0 & 1 \\ 1 & 0}$ and $B = \pmatrix{1 & 0 & 0\\ 0 & 0 & 1}$, we can take $X = \pmatrix{1 & 0 \\ 0 & 0}$. In general, I am not sure it is always possible.
Just use Gauss Jordan elimination. This gives you reduced row echelon form, which has a upper triangular-like nature to the result. You can also row swap to make it lower triangular if you'd like, after doing that.