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2.1. Elementary matrices

Definition 2.1.1.

There are three types of elementary row operations which can be performed on a matrix $A\in\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ . They are:

(i)

Multiplying a row by a non-zero number;
(ii)

Adding a multiple of one row to another row;
(iii)

Swapping two rows.

Example 2.1.2.

Let $A$ be the following matrix. This is how we will write row operations:

$A=\begin{pmatrix}1&2\\ 3&4\\ 5&6\end{pmatrix}\xrightarrow{R_{1}\leftrightarrow R_{2}}\begin{pmatrix}3&4\\ 1&2\\ 5&6\end{pmatrix}\xrightarrow{R_{3}=r_{3}+2r_{2}}\begin{pmatrix}3&4\\ 1&2\\ 7&10\end{pmatrix}.$

Here $R_{i}$ and $r_{i}$ are the names given to row number $i$ (counting from top to bottom). First we swapped $R_{1}$ and $R_{2}$ , and then we added $2\cdot r_{2}$ to $r_{3}$ . So $R_{i}=r_{i}+\lambda r_{j}$ means “The new row $i$ is the old row $i$ plus $\lambda$ times the old row $j$ ”. We will now explain this in more detail, by using elementary matrices.

Definition 2.1.3.

For any integers $n\geq 1$ and $1\leq i,j\leq n$ with $i\neq j$ , we define three kinds of elementary matrices in $\operatorname{M}_{{n}}({{\mathbb{R}}})$ as follows. We are assuming $\lambda\in{\mathbb{R}}$ is non-zero.

•

(Row multiplication):

$E_{i}(\lambda)=\begin{pmatrix}1&&&&&&\\ &\ddots&&&&\textup{\Huge 0}&\\ &&1&&&&\\ &&&\lambda&&&\\ &&&&1&&\\ &\textup{\Huge 0}&&&&\ddots&\\ &&&&&&1\\ \end{pmatrix}.$

If $A$ is an $n\times m$ matrix, then $E_{i}(\lambda)\cdot A$ (the matrix multiplication used here is from Definition 1.4.1) is the matrix $A$ except its $i$ -th row has been multiplied by the scalar $\lambda$ . We will write this row operation as $A\xrightarrow{R_{i}=\lambda r_{i}}E_{i}(\lambda)\cdot A$ .
•

(Row addition):

$E_{ij}(\lambda)=\begin{pmatrix}1&&&&&&\\ &\ddots&&&&\textup{\Huge 0}&\\ &&1&\cdots&\lambda&&\\ &&&\ddots&\vdots&&\\ &&&&1&&\\ &\textup{\Huge 0}&&&&\ddots&\\ &&&&&&1\\ \end{pmatrix},$

Here $\lambda$ occurs in the $i$ -th row and the $j$ -th column. If $A$ is an $n\times m$ matrix, then $E_{ij}(\lambda)\cdot A$ is the matrix whose $i$ -th row is the $i$ -th row of $A$ plus $\lambda$ times the $j$ -th row of $A$ . We will write this row operation as $A\xrightarrow{R_{i}=r_{i}+\lambda r_{j}}E_{ij}(\lambda)\cdot A$ .
•

(Row exchange):

$E_{ij}=\begin{pmatrix}1&&&&&&\\ &\ddots&&&&\textup{\Huge 0}&\\ &&0&\cdots&1&&\\ &&\vdots&\ddots&\vdots&&\\ &&1&\cdots&0&&\\ &\textup{\Huge 0}&&&&\ddots&\\ &&&&&&1\\ \end{pmatrix}.$

If $A$ is an $n\times m$ matrix, then $E_{ij}\cdot A$ is the matrix whose $i$ -th and $j$ -th rows have been exchanged. We will write this row operation as $A\xrightarrow{R_{i}\leftrightarrow R_{j}}E_{ij}\cdot A$ .

Example 2.1.4.

Consider the row operations in Example 2.1, where $n=3$ , because there are 3 rows. Let’s perform the same row operations by using elementary matrices. The first row operation is the same as multiplying $A$ on the left by $E_{12}$ . The second operation is the same as multiplying on the left by $E_{32}(2)$ . So multiply the following matrices:

$E_{32}(2)\left(E_{12}\cdot\begin{pmatrix}1&2\\ 3&4\\ 5&6\end{pmatrix}\right)=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&2&1\end{pmatrix}\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}1&2\\ 3&4\\ 5&6\end{pmatrix}=\begin{pmatrix}3&4\\ 1&2\\ 7&10\end{pmatrix}$

which is the same answer as we found in Example 2.1. Notice the order of the matrices. The last operation corresponds to the leftmost elementary matrix.

Remark 2.1.5.

One way of remembering how row operations correspond to elementary matrices, is as follows. The elementary matrix is the result of the row operation on the identity matrix $I_{n}$ , where

I_{n}=\left(\begin{array}[]{cccc}1\\ &\ddots&&\hbox{{\Huge 0}}\\ \hbox{{\Huge 0}}&&\ddots\\ &&&1\end{array}\right)=(\delta_{ij}),\quad\hbox{where}\quad\delta_{ij}=\left\{% \begin{array}[]{ll}1&\quad\hbox{if $i=j\;$,}\\ 0&\quad\hbox{otherwise.}\end{array}\right.

For us, the primary use of the elementary matrices will be to keep track of row operations on matrices. Notice the product of elementary matrices is the sequence of row operations written from right to left.

Remark 2.1.6.

Elementary column operations are defined similarly to the row operations, and we will use them in Section 4. The three kinds are the same as in Definition 2.1.1, except every instance of the word “row” is replaced with the word “column”.

Notation 2.1.7.

Given a matrix, we write $R_{i}$ or $r_{i}$ , ( $C_{i}$ or $c_{i}$ ) for the $i$ -th row ( $i$ -th column, respectively). During row operations, $r_{i}$ will often mean the “old row $i$ ”, and $R_{i}$ will often mean the “new row $i$ ”. For instance, if $A$ is the matrix

A=\begin{pmatrix}1&2&3\\ 4&5&6\end{pmatrix}\quad\hbox{then}\quad R_{1}=\big{(}1\quad 2\quad 3\big{)}% \quad\hbox{and}\quad C_{3}=\bigg{(}\begin{array}[]{r}3\\ 6\end{array}\bigg{)}.

Example 2.1.8.

Let $A=\begin{pmatrix}0&1&2\\ 1&-1&0\\ 0&-2&-1\end{pmatrix}\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ . Do the sequence of row operations:
- 1.
  
  add twice $R_{1}$ to $R_{3}$ , written $R_{3}=r_{3}+2r_{1}$ , then
- 2.
  
  swap $R_{2}$ and $R_{1}$ , written $R_{1}\leftrightarrow R_{2}$ , finally
- 3.
  
  multiply $R_{3}$ by $1/3$ , written $R_{3}=\frac{1}{3}r_{3}$ .
In terms of elementary matrices, this sequence of row operations is the same as the product

$E_{3}(1/3)\cdot E_{12}\cdot E_{31}(2)\cdot A=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&\frac{1}{3}\end{pmatrix}\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\ 0&1&0\\ 2&0&1\end{pmatrix}A=\begin{pmatrix}1&-1&0\\ 0&1&2\\ 0&0&1\end{pmatrix}.$

Notice that the sequence of elementary matrices is written in order starting on the right and ending on the left.