# 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 $\times$ 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. Let’s have a look on different types of elementary operations.

## 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 ith and jth rows are exchanged, it is shown by Ri ↔ Rj and if the ith and jth columns are exchanged, it is shown by Ci ↔ Cj.

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 ith row of a matrix by a non-zero number k, symbolically it can be denoted by RikRi. Similarly, for column it is given by CikCi.

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 ith row of a matrix to the jth row which is multiplied by a non-zero number k, symbolically it can be denoted by Ri ↔ Ri + kRj. Similarly, for column it is given by Ci ↔ Ci + kCj.

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:

$A = \begin{bmatrix} 1 & 2 & -3 \\ 8 & 3 & -6 \end{bmatrix}$<