Existence of Inverse Elementary Column Operation/Scalar Product of Column

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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 $\mathbf A' \in \map \MM {m, n}$.

\((\text {ECO} 1)\)   $:$   \(\ds \kappa_k \to \lambda \kappa_k \)    For some $\lambda \in K_{\ne 0}$, multiply column $k$ by $\lambda$      


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


Then $e'$ is the elementary column operation:

$e' := \kappa_k \to \dfrac 1 \lambda \kappa_k$


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:

$e := \kappa_k \to \lambda \kappa_k$

where $\lambda \ne 0$.

Then $\kappa'_k$ is such that:

$\forall a'_{k i} \in \kappa'_k: a'_{k i} = \lambda a_{k i}$


Now let $\map {e'} {\mathbf A'}$ be the elementary column operation which transforms $\mathbf A'$ to $\mathbf A$:

$e' := \kappa_k \to \dfrac 1 \lambda \kappa_k$

Because it is stipulated in the definition of an elementary column operation that $\lambda \ne 0$, it follows by definition of a field that $\dfrac 1 \lambda$ exists.

Hence $e'$ is defined.

So applying $e'$ to $\mathbf A'$ we get:

\(\ds \forall a_{i k} \in \kappa_k: \, \) \(\ds a_{i k}\) \(=\) \(\ds \dfrac 1 \lambda a'_{i k}\)
\(\ds \) \(=\) \(\ds \dfrac 1 \lambda \paren {\lambda a_{i k} }\)
\(\ds \) \(=\) \(\ds a_{i k}\)
\(\ds \leadsto \ \ \) \(\ds \forall a_{i k} \in \kappa_k: \, \) \(\ds a_{i k}\) \(=\) \(\ds a_{i k}\)
\(\ds \leadsto \ \ \) \(\ds \kappa_k\) \(=\) \(\ds \kappa_k\)
\(\ds \leadsto \ \ \) \(\ds \mathbf A\) \(=\) \(\ds \mathbf A\)

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

$\blacksquare$

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