Elementary Transformations and Equivalent matrices

Elementary Transformations and Equivalent matrices

Elementary transformations of a matrix

(i) Interchange of any two rows (or columns): R_i ↔ R _j (or C_i ↔C_j ) .

(ii) Multiplication of each element of a row (or column) by any non-zero scalar k : R_i → kR_i (or C_i →kC_i )

(iii) Addition to the elements of any row (or column) the same scalar multiples of corresponding elements of any other row (or column):

R_i → R_i +kR_j .(or C_i → C_i + kC_j )

Equivalent Matrices

Two matrices A and B are said to be equivalent if one is obtained from the another by applying a finite number of elementary transformations and we write it as A ~ B or B ~ A .