# Elementary Row Operations on Matrices

Elementary row operations on matrices are used to solve system of linear equations. Here we will see elementary row operations with illustrations.

## Elementary Row Operations on Matrices

There are three elementary row operations. Let us understand it one by one with example.

## Multiply a Row Through by a Nonzero Constant

In this elementary row operation we multiply every element of a particular row by a non zero constant $k$. Notation for this operation is given by $kR_{i}$. i.e.

$kR_{i}:=$ Multiply $i^{th}$ row by $k\ne 0$.

### Example 1

Let

 \begin{aligned} A&=\left[\begin{matrix}1&1&2&9\\0&2&-7&-16\\0&3&-11&-27\end{matrix}\right] \end{aligned}

Applying $\frac{1}{2}R_2$ (i.e. Multiply the second row by $1/2$) on the matrix $A$ we obtain

 \begin{aligned} B=\left[\begin{matrix}1&1&2&9\\0&1&-\frac{7}{2}&-8\\0&3&-11&-27\end{matrix}\right] \end{aligned}
Applying $5R_1$ (i.e. Multiply the first row by $5$ ) on the matrix $B$ we obtain

 \begin{aligned} C=\left[\begin{matrix}5&5&10&45\\0&1&-\frac{7}{2}&-8\\0&3&-11&-27\end{matrix}\right] \end{aligned}

## Interchange Two Rows

In this elementary row operation we interchange two rows.Notation for this operation is given by $R_{ij}$. i.e.

$R_{ij}:=$ Interchange $i^{th}$ and $j^{th}$ row.

### Example 2

Let

 \begin{aligned} A=\left[\begin{matrix}1&1&2&9\\0&2&-7&-16\\0&3&-11&-27\end{matrix}\right] \end{aligned}

Applying $R_{12}$ (i.e. Interchange the first and second rows ) on the matrix $A$ we obtain

 \begin{aligned} B=\left[\begin{matrix}0&2&-7&-16\\1&1&2&9\\0&3&-11&-27\end{matrix}\right] \end{aligned}

Applying $R_{13}$ (i.e. Interchange the first and third rows ) on the matrix $B$ we obtain

 \begin{aligned} C=\left[\begin{matrix}0&3&-11&-27\\1&1&2&9\\0&2&-7&-16\end{matrix}\right] \end{aligned}

## Add a Multiple of One Row to Another Row

In this elementary row operation we multiply every element of a row by a constant $k$ and then we add new obtained row's elements in to the corresponding elements of the another row.Notation for this operation is given by $R_{ij}(k)$. i.e.

$R_{ij}(k):=$ Multiply $i^{th}$ row by $k$ and add in to $j^{th}$ row.

### Example 3

Let

 \begin{aligned} A=\left[\begin{matrix}1&1&2&9\\0&2&-7&-16\\0&3&-11&-27\end{matrix}\right] \end{aligned}

Applying $R_{12(2)}$ (i.e. Multiply first row by 2 and add in to the second row ) on the matrix $A$ we obtain

 \begin{aligned} B&=\left[\begin{matrix}1&1&2&9\\0+2&2+2&-7+4&-16+18\\0&3&-11&-27\end{matrix}\right]\\ &=\left[\begin{matrix}1&1&2&9\\2&4&-3&2\\0&3&-11&-27\end{matrix}\right] \end{aligned}

Applying $R_{23}(-1)$ (i.e. Multiply second row by $-1$ and add in to the third row ) on the matrix $B$ we obtain

 \begin{aligned} C=\left[\begin{matrix}1&1&2&9\\2&4&-3&2\\-2&-1&-8&-29\end{matrix}\right] \end{aligned}

## Elementary Row Matrices

A matrix obtained by applying a single elementary row operation on an identity matrix is called an elementary row matrix.

Let us see some examples of elementary matrices.

### Example 4

Consider the $3\times 3$ identity matrix.

 \begin{aligned} I_3=\left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right] \end{aligned}

Let $E_1$ is obtained on applying $R_{12}$ on $I_3$. Then

 \begin{aligned} E_1=\left[\begin{matrix}0&1&0\\1&0&0\\0&0&1\end{matrix}\right] \end{aligned}

Let $E_2$ is obtained on applying $R_{12(2)}$ on $I_3$. Then

 \begin{aligned} E_2=\left[\begin{matrix}1&0&0\\2&1&0\\0&0&1\end{matrix}\right] \end{aligned}

Let $E_3$ is obtained on applying $5R_{3}$ on $I_3$. Then

 \begin{aligned} E_3=\left[\begin{matrix}1&0&0\\0&1&0\\0&0&5\end{matrix}\right] \end{aligned}

Here $E_1$, $E_2$, and $E_3$ are matrices obtained by applying a single elementary row operation on the identity matrix. So they are elementary row matrices.

Elementary row matrices are always non-singular (invertible).

## Left Multiplication by an Elementary Row Matrix

Let $A$ be a matrix. Then left multiplication by an elementary row matrix to the matrix $A$ produce the same matrix as you obtained by applying direct row operation (corresponding to elementary row matrix) on the matrix $A$. In short,

Left multiplication (pre-multiplication) by an elementary row matrix represents elementary row operation.

Let us understand above fact by following examples.

### Example 5

Starting with

$I=\left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right]$

Let $E$ be the elementary row matrix obtained by applying $R_{13}$ on $I$. Then

 \begin{aligned} E=\left[\begin{matrix}0&0&1\\0&1&0\\1&0&0\end{matrix}\right] \end{aligned}

Let

 \begin{aligned} A=\left[\begin{matrix}1&1&2&9\\2&4&-3&2\\0&3&-5&-3\end{matrix}\right] \end{aligned}

Now consider left multiplication (pre multiplication) to $A$ by $E$.

 \begin{aligned} EA&=\left[\begin{matrix}0&0&1\\0&1&0\\1&0&0\end{matrix}\right]\left[\begin{matrix}1&1&2&9\\2&4&-3&2\\0&3&-5&-3\end{matrix}\right]\\ &=\left[\begin{matrix}0&3&-5&-3\\2&4&-3&2\\1&1&2&9\end{matrix}\right] \end{aligned}

Observed that $EA$ is the same matrix obtained by applying $R_{13}$ on $A$.

### Example 6

Starting with $I=\left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right]$

Let $E$ be the elementary row matrix obtained by applying $2R_{2}$ on $I$. Then

 \begin{aligned} E=\left[\begin{matrix}1&0&0\\0&2&0\\0&0&1\end{matrix}\right] \end{aligned}

Let

 \begin{aligned} A=\left[\begin{matrix}1&1&2&9\\2&4&-3&2\\0&3&-5&-3\end{matrix}\right] \end{aligned}

Now consider left multiplication (pre multiplication) to $A$ by $E$.

 \begin{aligned} EA&=\left[\begin{matrix}1&0&0\\0&2&0\\0&0&1\end{matrix}\right]\left[\begin{matrix}1&1&2&9\\2&4&-3&2\\0&3&-5&-3\end{matrix}\right]\\ &=\left[\begin{matrix}1&1&2&9\\4&8&-6&4\\0&3&-5&-3\end{matrix}\right] \end{aligned}

Observed that $EA$ is the same matrix obtained by applying $2R_{2}$ on $A$.

### Example 7

Starting with $I=\left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right]$

Let $E$ be the elementary row matrix obtained by applying $R_{12}(3)$ on $I$. Then

 \begin{aligned} E=\left[\begin{matrix}1&0&0\\3&1&0\\0&0&1\end{matrix}\right] \end{aligned}

Let

 \begin{aligned} A=\left[\begin{matrix}1&1&2&9\\2&4&-3&2\\0&3&-5&-3\end{matrix}\right] \end{aligned}

Now consider left multiplication (pre multiplication) to $A$ by $E$.

 \begin{aligned} EA&=\left[\begin{matrix}1&0&0\\3&1&0\\0&0&1\end{matrix}\right]\left[\begin{matrix}1&1&2&9\\2&4&-3&2\\0&3&-5&-3\end{matrix}\right]\\ &=\left[\begin{matrix}1&1&2&9\\5&7&3&20\\0&3&-5&-3\end{matrix}\right] \end{aligned}

Observed that $EA$ is the same matrix obtained by applying $R_{12}(3)$ on $A$.

## Endnote

In this tutorial, you learned about how to perform elementary row operations on matrices. You also learned about elementary row operations with illustrated examples.