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.
To learn more about matrix algebra, please refer to the following tutorials:
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