# Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add

## Theorem

Let $\mathbf I$ denote the unit matrix of order $m$ over a field $K$.

Let $e$ be the elementary row operation acting on $\mathbf I$ as:

 $(\text {ERO} 2)$ $:$ $\ds r_i \to r_i + \lambda r_j$ For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$

for $1 \le i \le m$, $1 \le j \le m$.

Let $\mathbf E$ be the elementary row matrix defined as:

$\mathbf E = e \paren {\mathbf I}$

$\mathbf E$ is the square matrix of order $m$ of the form:

$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{a i} \cdot \delta_{j b}$

where:

$E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$
$\delta_{a b}$ is the Kronecker delta:
$\delta_{a b} = \begin {cases} 1 & : \text {if$a = b$} \\ 0 & : \text {if$a \ne b$} \end {cases}$

## Proof

By definition of the unit matrix:

$I_{a b} = \delta_{a b}$

where:

$I_{a b}$ denotes the element of $\mathbf I$ whose indices are $\tuple {a, b}$.

By definition, $\mathbf E$ is the square matrix of order $m$ formed by applying $e$ to the unit matrix $\mathbf I$.

That is, all elements of row $i$ of $\mathbf I$ are to have the corresponding elements of row $j$ added to them after the latter have been multiplied by $\lambda$.

By definition of unit matrix:

all elements of row $i$ are $0$ except for element $I_{i i}$, which is $1$.
all elements of row $j$ are $0$ except for element $I_{j j}$, which is $1$.

Thus in $\mathbf E$:

where $a \ne i$, $E_{a b} = \delta_{a b}$
where $a = i$:
$E_{a b} = \delta_{a b}$ where $b \ne j$
$E_{a b} = \delta_{a b} + \lambda \cdot 1$ where $b = j$

That is:

$E_{a b} = \delta_{a b}$ for all elements of $\mathbf E$

except where $a = i$ and $b = j$, at which element:

$E_{a b} = \delta_{a b} + \lambda$

That is:

$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{a i} \cdot \delta_{j b}$

Hence the result.

$\blacksquare$

## Also presented as

This can also be seen presented as:

$E_{a b} = \begin{cases} \delta_{a b} & : a \ne i \\ \delta_{i b} + \lambda \delta_{j b} & : a = i \end{cases}$

when it is considered desirable to make the nature of the rows more easily understood.