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 $\mathbf A = \mathbf A$.
$\text {ECO} 1$: Scalar Product of Column
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.
$\Box$
$\text {ECO} 2$: Add Scalar Product of Column to Another
Let $\map e {\mathbf A}$ be the elementary column operation:
- $e := \kappa_k \to \kappa_k + \lambda r_l$
Then $\kappa'_k$ is such that:
- $\forall a'_{i k} \in \kappa'_k: a'_{i k} = a_{i k} + \lambda a_{i l}$
Now let $\map {e'} {\mathbf A'}$ be the elementary column operation which transforms $\mathbf A'$ to $\mathbf A$:
- $e' := \kappa'_k \to \kappa'_k - \lambda \kappa'_l$
Applying $e'$ to $\mathbf A'$ we get:
| \(\ds \forall a_{i k} \in \kappa_k: \, \) | \(\ds a_{i k}\) | \(=\) | \(\ds a'_{i k} - \lambda a'_{i l}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a_{i k} + \lambda a_{i l} } - \lambda a'_{i l}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a_{i k} + \lambda a_{i l} } - \lambda a_{i l}\) | as $\lambda a'_{i l} = \lambda a_{i l}$: column $l$ was not changed by $e$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds a_{i k}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall a_{i k} \in \kappa_k: \, \) | \(\ds a_{i k}\) | \(=\) | \(\ds a_{i k}\) | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \kappa_{i k}\) | \(=\) | \(\ds \kappa_{i k}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \mathbf A\) | \(=\) | \(\ds \mathbf A\) |
It is noted that for $e'$ to be an elementary column operation, the only possibility is for it to be as defined.
$\Box$
$\text {ECO} 3$: Exchange Columns
Let $\map e {\mathbf A}$ be the elementary column operation:
- $e := \kappa_k \leftrightarrow \kappa_l$
Thus we have:
| \(\ds \kappa'_k\) | \(=\) | \(\ds \kappa_l\) | ||||||||||||
| \(\, \ds \text {and} \, \) | \(\ds \kappa'_l\) | \(=\) | \(\ds \kappa_k\) |
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:
| \(\ds \kappa_k\) | \(=\) | \(\ds \kappa'_l\) | ||||||||||||
| \(\, \ds \text {and} \, \) | \(\ds \kappa_l\) | \(=\) | \(\ds \kappa'_k\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \kappa_k\) | \(=\) | \(\ds \kappa_k\) | |||||||||||
| \(\, \ds \text {and} \, \) | \(\ds \kappa_l\) | \(=\) | \(\ds \kappa_l\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \mathbf A\) | \(=\) | \(\ds \mathbf A\) |
It is noted that for $e'$ to be an elementary column operation, the only possibility is for it to be as defined.
$\Box$
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.
$\blacksquare$