Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse

Theorem
Let $e$ be an elementary column operation.

Let $\mathbf E$ be the elementary column matrix corresponding to $e$.

Let $e'$ be the inverse of $e$.

Then the elementary column matrix corresponding to $e'$ is the inverse of $\mathbf E$.

Also see

 * Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse