Existence of Inverse Elementary Row Operation/Add Scalar Product of Row to Another

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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 row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.

\((\text {ERO} 2)\)   $:$   \(\ds r_k \to r_k + \lambda r_l \)    For some $\lambda \in K$, add $\lambda$ times row $k$ to row $l$      


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


Then $e'$ is the elementary row operation:

$e' := r'_k \to r'_k - \lambda r'_l$


Proof

In the below, let:

$r_k$ denote row $k$ of $\mathbf A$
$r'_k$ denote row $k$ of $\mathbf A'$
$r''_k$ denote row $k$ of $\mathbf A''$

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


By definition of elementary row operation:

only the row or rows directly operated on by $e$ is or are different between $\mathbf A$ and $\mathbf A'$

and similarly:

only the row or rows 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 rows directly affected will be under consideration when showing that $\mathbf A = \mathbf A''$.


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

$e := r_k \to r_k + \lambda r_l$

Then $r'_k$ is such that:

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


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

$e' := r'_k \to r'_k - \lambda r'_l$


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

\(\ds \forall a''_{k i} \in r''_k: \, \) \(\ds a''_{k i}\) \(=\) \(\ds a'_{k i} - \lambda a'_{l i}\)
\(\ds \) \(=\) \(\ds \paren {a_{k i} + \lambda a_{l i} } - \lambda a'_{l i}\)
\(\ds \) \(=\) \(\ds \paren {a_{k i} + \lambda a_{l i} } - \lambda a_{l i}\) as $\lambda a'_{l i} = \lambda a_{l i}$: row $l$ was not changed by $e$
\(\ds \) \(=\) \(\ds a_{k i}\)
\(\ds \leadsto \ \ \) \(\ds \forall a''_{k i} \in r''_k: \, \) \(\ds a''_{k i}\) \(=\) \(\ds a_{k i}\)
\(\ds \leadsto \ \ \) \(\ds r''_{k i}\) \(=\) \(\ds r_{k i}\)
\(\ds \leadsto \ \ \) \(\ds \mathbf A''\) \(=\) \(\ds \mathbf A\)

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

$\blacksquare$