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Elementary Operations of a Matrix

The following three operations applied on the rows (columns) of a matrix are called elementary row (column) transformations.

(i) Interchange of any two rows (columns)

If ith row (column) of a matrix is interchanged with the jth row (column), it will be denoted by R_i ↔ R_j (C_i ↔ C_j).

For example: A = matrix22 , then by applying R₁ → R₂ we get B = matrix23 .

(ii) Multiplying all elements of a row (column) of a matrix by a non-zero scalar. If the elements of ith row (column) are multiplied by non-zero scalar k, it will be denoted by R_l→R_i (k) [C_i→C_i (k)] or R_l→kR_i [C_i→kC_i].

If A = matrix24 , then by applying R₂→ 3R₂, we obtain B = matrix25 .

(iii) Adding to the elements of a row (column), the corresponding elements of any other row (column) multiplied by any scalar k.

If k times the elements of jth row (column) are added to the corresponding elements of the ith row (column), it will be denoted by R_i→R_i + k R_j (C_i→C_i + kC_j).

If A= matrix26 , then application of elementary operation R₃→R₃ + 2R₁ lead to B = matrix27 .

Illustration:

Using elementary row transformations and find the inverse of the matrix A= matrix28 .

Solution:

matrix29

matrix30

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