. . Row-Echelon form Using the row elementary operations, we can transform a given non-zero matrix to a simplified form called a Row-echelon form . In a row-echelon form, we may have rows all of whose entries are zero.
Such rows are called zero rows . A non-zero row is one in which at least one of the entries is not zero. For instance, in the matrix , R are non-zero rows and R is a zero row. Definition .
A non-zero matrix E is said to be in a row-echelon form if: (i) All zero rows of E occur below every non-zero row of E . (ii) The first non-zero element in any row i of E occurs in the j th column of E , then all other entries in the j th column of E below the first non-zero element of row i are zeros. (iii) The first non-zero entry in the i th row of E lies to the left of the first non-zero entry in i + th row of E . - - Note A non-zero matrix is in a row-echelon form if all zero rows occur as bottom rows of the matrix, and if the first non-zero element in any lower row occurs to the right of the first non- zero entry in the higher row.
The following matrices are in row-echelon form:(i) ,(ii) Consider the matrix in (i). Go up row by row from the last row of the matrix. The third row is a zero row. The first non-zero entry in the second row occurs in the third column and it lies to the right of the first non-zero entry in the first row which occurs in the second column.
So the matrix is in row- echelon form. Consider the matrix in (ii). Go up row by row from the last row of the matrix. All the rows are non-zero rows.
The first non-zero entry in the third row occurs in the fourth column and it occurs to the right of the first non-zero entry in