. . 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.
- - Applications of Matrices and Determinants 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 ↔ (ii) The multiplication of each element of i th row by a non-zero constant λ is denoted by R →λ (iii) Addition to i th row, a non-zero constant λ multiple of j th row is denoted by R + λ Similar notations are used for elementary column transformations. Definition . 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 B to mean that the matrix A is equivalent to the matrix B . For instance, let us consider a matrix A = After performing the elementary row operation R 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 A B = 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.