# Elementary Column 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 Column Operations On Matrices

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

## Multiply a Column Through by a Nonzero Constant

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

$kC_{i}:=$ Multiply $i^{th}$ column 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 $\frac{1}{2}C_2$ (i.e. Multiply the second row by $1/2$ ) on the matrix $A$ we obtain

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

Applying $(-3)R_3$ (i.e. Multiply the second column by $-3$ ) on the matrix $B$ we obtain

 \begin{aligned} C&=\left[\begin{matrix}1&\dfrac{1}{2}&-6&9\\0&1&21&-16\\0&\dfrac{3}{2}&33&-27\end{matrix}\right] \end{aligned}

## Interchange Two Columns

In this elementary column operation we interchange two columns.Notation for this operation is given by $C_{ij}$. i.e.

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

### 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 $C_{12}$ (i.e. Interchange the first and second columns ) on the matrix $A$ we obtain

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

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

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

## Add a Multiple of One Row to Another Row

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

$C_{ij}(k):=$ 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 $C_{12(2)}$ (i.e. Multiply first column by 2 and add in to the second column ) on the matrix $A$ we obtain

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

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

 \begin{aligned} C&=\left[\begin{matrix}1&3&-1&9\\0&2&-9&-16\\0&3&-14&-27\end{matrix}\right] \end{aligned}

## Elementary Column Matrices

A matrix obtained by applying a single elementary column operation on an identity matrix is called an elementary column 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 $C_{12}$ 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 $C_{12(2)}$ on $I_3$. Then

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

Let $E_3$ is obtained on applying $5C_{3}$ 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 column operation on the identity matrix. So they are elementary column matrices.

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

## Right Multiplication by an Elementary Column Matrix

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

Right multiplication (post-multiplication) by an elementary column matrix represents elementary column 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 column matrix obtained by applying $C_{13}$ 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\\2&4&-3\\0&3&-5\end{matrix}\right] \end{aligned}

Now consider right multiplication (post multiplication) to $A$ by $E$.

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

Observed that $AE$ is the same matrix obtained by applying $C_{13}$ 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 column matrix obtained by applying $2C_{2}$ 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\\2&4&-3\\0&3&-5\end{matrix}\right] \end{aligned}

Now consider right multiplication (post multiplication) to $A$ by $E$.

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

Observed that $EA$ is the same matrix obtained by applying $2C_{2}$ 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 column matrix obtained by applying $C_{12}(3)$ on $I$. Then

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

Let

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

Now consider right multiplication (post multiplication) to $A$ by $E$.

 \begin{aligned} AE&=\left[\begin{matrix}1&1&2\\2&4&-3\\0&3&-5\end{matrix}\right]\left[\begin{matrix}1&3&0\\0&1&0\\0&0&1\end{matrix}\right]\\ &=\left[\begin{matrix}1&4&2\\2&10&-3\\0&3&-5\end{matrix}\right] \end{aligned}

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

## Endnote

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