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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

\(\begin{array}{l}\times\end{array} \)
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 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:

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

We apply 

\(\begin{array}{l}R_{1}\leftrightarrow R_{2}\end{array} \)
and obtain:

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

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 ith row of a matrix by a non-zero number k, symbolically it can be denoted by Ri → kRi. Similarly, for column it is given by Ci → kCi.

For example, given the matrix A below:

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

We apply R1→3R1 and obtain:

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

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 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:

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

We apply R2→R2+4Rand obtain:

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

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Frequently Asked Questions on Elementary Operation of Matrix

What is meant by the elementary operation of matrix?

The elementary operation of a matrix, also known as elementary transformation are the operations performed on rows and columns of a matrix to transform the given matrix into a different form inorder to make the calculation simpler.

Mention the different types of elementary operations of a matrix?

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

Can we interchange rows in a matrix?

Yes, we can interchange the rows of a matrix to get a new matrix. For example, R1↔R2 or R1↔R3 and so on.

Does the elementary operation of the matrix affect the solution of the system of linear equations?

No, the elementary operation of the matrix does not affect the solution of the system of linear equations.

Why do we use elementary row operations?

Elementary row operations are used in Gaussian elimination in order to transform the given matrix into the reduced row Echelon form.

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