Understanding Operation Transformation

Introduction:
              Elementary operation transformation of matrix is defined as

(i) Exchange any two rows (or columns)
(ii) Multiply the every element of a row (or column) by any non-zero scalar.
(iii) Addition to the elements of any row (or column) the same scalar multiples of corresponding elements of any other row (or column).
    The above statement of transformation matrix is represented by symbols as follows:
(i)                 Ri ↔ Rj (Ci ↔ Cj) ;
(ii)               Ri → kRi (Ci → kCi)
(iii)             Ri → Ri + kRj (Ci → Ci + kCj)
Elementary Operation(transformation)of Matrix:

The blown result is used in elementary operation (transformation) of matrix.
Equivalent matrices  and diagonal matrix have the same rank
Echelon form of a matrix:
A matrix A (of order m × n) is said to be in echelon form (triangular form) if
(i) All rows of A which has every entries 0 occurs under every row which has a non-zero entry.
(ii) The first non-zero entry in each non-zero row is 1.
(iii) The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row.
By elementary operations one can easily bring the given matrix to the echelon form.
The rank of a matrix in echelon form is equal to number of non-zero rows of the matrix.



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