Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse/Proof 2

== Theorem ==
== Proof ==
We will demonstrate this for each of the $3$ types of elementary row operation.
In the below:
:$e$ denotes a given elementary row operation
:$\mathbf E$ denotes the elementary row matrix corresponding to $e$
:$e'$ denotes the inverse of $e$
:$\mathbf E'$ denotes the elementary row matrix corresponding to $e'$.
Let $n$ denote the order of $\mathbf E$ and $\mathbf E'$.
The strategy is to demonstrate that:
:$\mathbf E \mathbf E' = \mathbf I$
where $\mathbf I$ denotes the unit matrix of order $n$.
Let $x_{i, j}$ and $y_{i, j}$ denote the elements of $\mathbf E$ and $\mathbf E'$ respectively at indices $\tuple {i, j}$.
Let $z_{i j}$ denote the element of $\mathbf E \mathbf E'$ at indices $\tuple {i, j}$.
=== $\text {ERO} 1$: Scalar Product of Row ===
Let $e$ be the elementary row operation:
:$e := r_k \to \lambda r_k$
where $\lambda \ne 0$.
From Elementary Matrix corresponding to Elementary Row Operation, $\mathbf E$ is of the form:
:$x_{a b} = \begin {cases} \delta_{a b} & : a \ne k \\ \lambda \cdot \delta_{a b} & : a = k \end {cases}$
where:
:$\delta_{a b}$ is the Kronecker delta:
::$\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$
From Existence of Inverse Elementary Row Operation: Scalar Product of Row, $e'$ is the elementary row operation:
:$e' := r_k \to \dfrac 1 \lambda r_k$
From Elementary Matrix corresponding to Elementary Row Operation, $\mathbf E'$ is of the form:
:$y_{a b} = \begin {cases} \delta_{a b} & : a \ne k \\ \dfrac 1 \lambda \cdot \delta_{a b} & : a = k \end {cases}$
By definition of matrix product:
:$\ds \forall a, b \in \set {1, 2, \ldots, n}: z_{a b} = \sum_{p \mathop = 1}^n x_{a p} y_{p b}$
Thus $z_{a b} \ne 0$ $a = p$ and $b = p$.
When $a \ne k$:
:$x_{a a} = y_{a a} = 1$
and so:
:$z_{a a} = 1 \times 1 = 1$
When $a = k$:
:$x_{a a} = \lambda$, $y_{l b} = \dfrac 1 \lambda$
and so:
:$z_{a a} = \lambda \times \dfrac 1 \lambda = 1$
and for all $z_{a b}$ where $a \ne b$:
:$z_{a b} = 0$
That is:
:$z_{a b} = \delta_{a b}$
and by definition:
:$\mathbf E \mathbf E' = \mathbf I$
=== $\text {ERO} 2$: Add Scalar Product of Row to Another ===
Let $e$ be the elementary row operation:
:$e := r_i \to r_i + \lambda r_j$
From Elementary Matrix corresponding to Elementary Row Operation, $\mathbf E$ is of the form:
:$x_{a b} = \delta_{a b} + \lambda \cdot \delta_{a i} \cdot \delta_{j b}$
From Existence of Inverse Elementary Row Operation: Add Scalar Product of Row to Another, $e'$ is the elementary row operation:
:$e' := r_i \to r_i - \lambda r_j$
From Elementary Matrix corresponding to Elementary Row Operation, $\mathbf E'$ is of the form:
:$y_{a b} = \delta_{a b} - \lambda \cdot \delta_{a i} \cdot \delta_{j b}$
We have that:
But:
:$\lambda \cdot \delta_{j b} \cdot \delta_{a i} - \lambda \cdot \delta_{j b} \cdot \delta_{a i} = 0$
So everything vanishes except $\delta_{a b}$, and so:
:$z_{a b} = \delta_{a b}$
and by definition, again:
:$\mathbf E \mathbf E' = \mathbf I$
=== $\text {ERO} 3$: Exchange Rows ===
Let $e$ be the elementary row operation:
:$e := r_i \leftrightarrow r_j$
From Elementary Matrix corresponding to Elementary Row Operation, $\mathbf E$ is of the form:
:$x_{a b} = \begin {cases} \delta_{a b} & : \text {if $a \ne i$ and $a \ne j$} \\ \delta_{j b} & : \text {if $a = i$} \\ \delta_{i b} & : \text {if $a = j$} \end {cases}$
From Existence of Inverse Elementary Row Operation: Exchange Rows, $e'$ is the elementary row operation:
:$e' := r_i \leftrightarrow r_j$
From Elementary Matrix corresponding to Elementary Row Operation, $\mathbf E'$ is of the form:
:$y_{a b} = \begin {cases} \delta_{a b} & : \text {if $a \ne i$ and $a \ne j$} \\ \delta_{j b} & : \text {if $a = i$} \\ \delta_{i b} & : \text {if $a = j$} \end {cases}$
By definition of matrix product:
:$\ds \forall a, b \in \set {1, 2, \ldots, n}: z_{a b} = \sum_{p \mathop = 1}^n x_{a p} y_{p b}$
When $a \ne i$ and $b \ne j$ we have:
When $a = i$ and $b \ne i$ we have:
When $a = j$ and $b \ne j$ we have:
When $a = b = i$ we have:
When $a = b = j$ we have:
Hence by definition, again:
:$\mathbf E \mathbf E' = \mathbf I$
Thus in all cases:
:$\mathbf E \mathbf E' = \mathbf I$
Hence the result.
== Sources ==
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: Exercises: $1.15 \ \text {(a)}$
Category:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse