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Elementary transformation of a matrix

by krishna

When we apply any of the following operation on a matrix, then we call it an elementary transformation.

1. Interchanging any two rows (or columns). This transformation is indicated by , if the ith and jth rows are interchanged.

2. Multiplication of the elements of any row   (or column) by a non-zero scalar quantity k is denoted by (k.).

3. Addition of constant multiplication of the elements of any row  to the corresponding elements of any other row Rj is denoted by ( + k).

If a matrix B is obtained from a matrix A by one or more E-operations, then B will be equivalent to A. We use the symbol ~ for equivalence.

i.e., A ~ B.

Inverse of a matrix by using el. transformation-

The following transformations are elementary transformations-

1. Interchange of any two rows (column)

2. Multiplication of any row or column by any non-zero scalar quantity k.

3. Addition to one row (column) of another row(column) multiplied by any non-zero scalar.

The symbol ~ we use for equivalence.

Elementary matrices

If we get a square matrix from an identity or unit matrix by using any single el. transformation, we call it an elementary matrix.

Note- Every elementary row transformation of a matrix can affect by pre-multiplication with the corresponding elementary matrix.

The method of finding inverse of a non-singular matrix by using el. transformation-

Working steps-

1. Write A = IA

2. Perform an elementary row transformation of A on the left side and I on the right side.

3. Apply elementary row transformation until ‘A’ (left side) reduces to I, then I reduces to .

Example-1: Find the inverse of matrix ‘A’ by using elementary transformation

Sol.  Write the matrix ‘A’ as-  

A = IA

Now apply the following operation

So that

Example-2: Find the inverse of matrix ‘A’ by using el. transformation-

Sol.  Write the given matrix ‘A’ as-  

hence A = IA

So that

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