Exercise 3.1.6

If $B$ can be obtained from $A$ by an elementary row operation, we could write $B=\mathit{EA}$. So we have $B^t=A^t{E}^t$ and this means $B$ can be obtained by $A$ by elementary column operation with corresponding elementary matrix $E^t$. If $B$ can be obtained from $A$ by an elementary column operation, we could write $B=\mathit{AE}$. So we have $B^t=E^t{A}^t$ and this means $B$ can be obtained by $A$ by elementary row operation with corresponding elementary matrix $E^t$.