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Definition, Theorem | Elementary row and column operations - Elementary Transformations of a Matrix | 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

Chapter: 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

Elementary Transformations of a Matrix

A matrix can be transformed to another matrix by certain operations called elementary row operations and elementary column operations.

Elementary Transformations of a Matrix

A matrix can be transformed to another matrix by certain operations called elementary row operations and elementary column operations.

 

Elementary row and column operations

Elementary row (column) operations on a matrix are as follows:

(i) The interchanging of any two rows (columns) of the matrix

(ii) Replacing a row (column) of the matrix by a non-zero scalar multiple of the row (column) by a non-zero scalar.

(iii) Replacing a row (column) of the matrix by a sum of the row (column) with a non-zero scalar multiple of another row (column) of the matrix.

Elementary row operations and elementary column operations on a matrix are known as elementary transformations.

We use the following notations for elementary row transformations:

i.   Interchanging of ith and jth rows is denoted by Ri Rj .

ii. The multiplication of each element of ith row by a non-zero constant λ is denoted by Ri   λ Ri .

iii. Addition to ith row, a non-zero constant λ multiple of jth row is denoted by Ri Ri + λ Rj . Similar notations are used for elementary column transformations.

 

Definition 1.4

Two matrices A and B of same order are said to be equivalent to one another if one can be obtained from the other by the applications of elementary transformations. Symbolically, we write A ~ B to mean that the matrix A is equivalent to the matrix B .

 

For instance, let us consider a matrix 

After performing the elementary row operation R2 R2 + R1 on A , we get a matrix B in which the second row is the sum of the second row in A and the first row in A .

Thus, we get 

The above elementary row transformation is also represented as follows:


Note

An elementary transformation transforms a given matrix into another matrix which need not be equal to the given matrix.

 

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants


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