Existence of Inverse Elementary Column Operation

Theorem
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$.

Let $\mathbf A \in \map \MM {m, n}$ be a matrix.

Let $\map e {\mathbf A}$ be an elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.

Let $\map {e'} {\mathbf A'}$ be the inverse of $e$.

Then $e'$ is an elementary column operation which always exists and is unique.

Proof
Let us take each type of elementary column operation in turn.

For each $\map e {\mathbf A}$, we will construct $\map {e'} {\mathbf A'}$ which will transform $\mathbf A'$ into a new matrix $\mathbf A'' \in \map \MM {m, n}$, which will then be demonstrated to equal $\mathbf A$.

In the below, let:
 * $\kappa_k$ denote column $k$ of $\mathbf A$
 * $\kappa'_k$ denote column $k$ of $\mathbf A'$
 * $\kappa_k$ denote column $k$ of $\mathbf A$

for arbitrary $k$ such that $1 \le k \le m$.

By definition of elementary column operation:
 * only the column or columns directly operated on by $e$ is or are different between $\mathbf A$ and $\mathbf A'$

and similarly:
 * only the column or columns directly operated on by $e'$ is or are different between $\mathbf A'$ and $\mathbf A''$.

Hence it is understood that in the following, only those columns directly affected will be under consideration when showing that $\mathbf A = \mathbf A''$.

$\text {ECO} 3$: Exchange Columns
Thus in all cases, for each elementary column operation which transforms $\mathbf A$ to $\mathbf A'$, we have constructed the only possible elementary column operation which transforms $\mathbf A'$ to $\mathbf A$.

Hence the result.