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.
To learn more about matrix algebra, please refer to the following tutorials:
Let me know in the comments if you have any questions on Elementary column operations on matrices and your thought on this article.