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 [latex]\times[/latex] 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 a matrix. The elementary operations or transformation of a matrix are the operations performed on rows and columns of a matrix to transform the given matrix into a different form in order to make the calculation simpler. In this article, we are going to learn three basic elementary operations of matrix in detail with examples.

Three Basic Elementary Operations of Matrix

We know that elementary row operations are the operations that are performed on rows of a matrix. Similarly, elementary column operations are the operations that are performed on columns of a matrix.

The three basic elementary operations or transformation of a matrix are:

1. Interchange of any two rows or two columns.
2. Multiplication of row or column by a non-zero number.
3. Multiplication of row or column by a non-zero number and add the result to the other row or column.

Now, let us discuss these three basic elementary operations of a matrix in detail.

Case 1: Interchange of any Two Rows or Two Columns

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:

[latex]A = \begin{bmatrix} 1 & 2 & -3 \\ 4 & -5 & 6 \end{bmatrix}[/latex]

We apply [latex]R_{1}\leftrightarrow R_{2}[/latex] and obtain:

[latex]A = \begin{bmatrix} 4 & -5 & 6 \\ 1 & 2 & -3 \end{bmatrix}[/latex]

Case 2: Multiplication of Row or Column by a Non-zero Number

The elements of any row (or column) of a matrix can be multiplied by 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 → kR_i. Similarly, for column it is given by C_i → kC_i.

For example, given the matrix A below:

[latex]A = \begin{bmatrix} 1 & 2 & -3 \\ 4 & -5 & 6 \end{bmatrix}[/latex]

We apply R₁→3R₁ and obtain:

[latex]A = \begin{bmatrix} 3 & 6 & -9 \\ 4 & -5 & 6 \end{bmatrix}[/latex]

Case 3: Multiplication of Row or Column by a Non-zero Number and Add the Result to the Other Row or Column

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 + kR_j. Similarly, for column it is given by C_i → C_i + kC_j.

For example, given the matrix A below:

[latex]A = \begin{bmatrix} 1 & 2 & -3 \\ 4 & -5 & 6 \end{bmatrix}[/latex]

We apply R₂→R₂+4R₁ and obtain:

[latex]A = \begin{bmatrix} 1 & 2 & -3 \\ 8 & 3 & -6 \end{bmatrix}[/latex]

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