# Elementary Operation of Matrix

A matrix is an array of numbers arranged in the form of rows and columns. The number of rows and columns of a matrix are known as its dimensions which is given by ** m \(\times\) n,** where

*m*and

*n*represent the number of rows and columns respectively. Apart from basic mathematical operations there are certain elementary operations that can be performed on matrix namely transformations. It is a special type of matrix that can be illustrate 2d and 3d transformations.

## Types of Elementary Operations

There are two types of elementary operations of a matrix:

**Elementary row operations:**when they are performed on rows of a matrix.**Elementary column operations:**when they are performed on columns of a matrix.

### Elementary Operations of a Matrix

- Any 2 columns (or rows) of a matrix can be exchanged. If the i
^{th}and j^{th}rows are exchanged, it is shown by R_{i}↔ R_{j}and if the i^{th}and j^{th}columns are exchanged, it is shown by C_{i}↔ C_{j}.

For example, given the matrix A below:

\(A = \begin{bmatrix} 1 & 2 & -3 \\ 4 & -5 & 6 \end{bmatrix}\)

We apply \(R_{1}\leftrightarrow R_{2}\) and obtain:

\(A = \begin{bmatrix} 4 & -5 & 6 \\ 1 & 2 & -3 \end{bmatrix}\)

- The elements of any row (or column) of a matrix can be multiplied with a non-zero number. So if we multiply the i
^{th}row of a matrix by a non-zero number*k*, symbolically it can be denoted by R_{i}↔*k*R_{i}. Similarly, for column it is given by C_{i}↔*k*C_{i}.

For example, given the matrix A below:

\(A = \begin{bmatrix} 1 & 2 & -3 \\ 4 & -5 & 6 \end{bmatrix}\)

We apply \(R_{1}\leftrightarrow 3R_{1}\) and obtain:

\(A = \begin{bmatrix} 3 & 6 & -9 \\ 4 & -5 & 6 \end{bmatrix}\)

- The elements of any row (or column) can be added with the corresponding elements of another row (or column) which is multiplied by a non-zero number. So if we add the i
^{th}row of a matrix to the j^{th}row which is multiplied by a non-zero number*k*, symbolically it can be denoted by R_{i}↔ R_{i}+*k*R_{j}. Similarly, for column it is given by C_{i}↔ C_{i}+*k*C_{j}.

For example, given the matrix A below:

\(A = \begin{bmatrix} 1 & 2 & -3 \\ 4 & -5 & 6 \end{bmatrix}\)

We apply \(R_{2}\leftrightarrow R_{2}+4R_{1}\) and obtain:

\(A = \begin{bmatrix} 1 & 2 & -3 \\ 8 & 3 & -6 \end{bmatrix}\)