Multiplications by unimodular matrices

Question

I feel like this must have an obvious answer, but my knowledge of integer arithmetic is limited.

Given an (integer) matrix $A$ of dimension $m \times n$ and an unimodular matrix $U_l$ of dimension $m \times m$, does there always exist an unimodular matrix $U_r$ of dimension $n \times n$ such that $U_l \cdot A$ = $A \cdot U_r$ ?

If so, what is an efficient way to compute $U_r$ from $U_l$ (or the other way round) ?

Answer 1

No. If $n=1$ and $m\ne1$, then $AU_r=\pm A$, whereas $U_lA$ isn't necessarily $\pm A$.