A matrix is said to be in "'reduced row echelon form "'if furthermore all of the leading coefficients are equal to 1 ( which can be achieved by using the elementary row operation of type 2 ), and in every column containing a leading coefficient, all of the other entries in that column are zero ( which can be achieved by using elementary row operations of type 3 ).
32.
A matrix is said to be in "'reduced row echelon form "'if furthermore all of the leading coefficients are equal to 1 ( which can be achieved by using the elementary row operation of type 2 ), and in every column containing a leading coefficient, all of the other entries in that column are zero ( which can be achieved by using elementary row operations of type 3 ).
33.
For any index i for which \ alpha _ i \ nmid \ alpha _ { i + 1 }, one can repair this shortcoming by operations on rows and columns i and i + 1 only : first add column i + 1 to column i to get an entry \ alpha _ { i + 1 } in column " i " without disturbing the entry \ alpha _ i at position ( i, i ), and then apply a row operation to make the entry at position ( i, i ) equal to \ beta = \ gcd ( \ alpha _ i, \ alpha _ { i + 1 } ) as in Step II; finally proceed as in Step III to make the matrix diagonal again.
