\begin{definition}[Definition:Inverse of Elementary Column Operation]
Let $e$ be an elementary column operation which transforms a matrix $\mathbf A$ to another matrix $\mathbf B$.
Let $e'$ be an elementary column operation which transforms $\mathbf B$ back to $\mathbf A$.
Then $e'$ is the '''inverse of the elementary column operation $e$'''.
\end{definition}