Let $A,B,C\in M_{(n\times k)} [\mathbb Z]$. In which conditions we can achieve $AC^T$ from $AB^T$ just by interchanging some appropriate columns (or rows) with each other ?!
(For example if we interchange column 2 with column 5 of $AB^T$ we can have $AC^T$.)
Note that $M_{(n\times k)} [\mathbb Z]$ is the set of $n\times k$ matrices with entries in integers.