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 (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 *i*^{th} and *j*^{th}
rows is denoted by *R _{i} *â†”

ii. The multiplication of each element of *i*^{th} row
by a non-zero constant *Î»** *is denoted by *R _{i} *â†’

iii. Addition to *i*^{th} row, a non-zero constant *Î» *multiple
of *j*^{th} row is denoted by *R _{i} *â†’

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 *R*_{2}
â†’ *R*_{2} + *R*_{1}
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:

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

Tags : Definition, Theorem | Elementary row and column operations , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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

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