Existence of Inverse Elementary Column Operation/Exchange Columns

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 the 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 the elementary column operation:


 * $e' := \kappa_k \leftrightarrow \kappa_l$

That is:
 * $e' = e$

Proof
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 $1 \le k \le n$.

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 $\mathbf A = \mathbf A''$.

Let $\map e {\mathbf A}$ be the elementary column operation:


 * $e := \kappa_k \leftrightarrow \kappa_l$

Thus we have:

Now let $\map {e'} {\mathbf A'}$ be the elementary column operation which transforms $\mathbf A'$ to $\mathbf A''$:
 * $e' := \kappa'_k \leftrightarrow \kappa'_l$

Applying $e'$ to $\mathbf A'$ we get:

It is noted that for $e'$ to be an elementary column operation, the only possibility is for it to be as defined.