Suppose that a row of a matrix is called zero if all its entries are zero. Then a matrix is in reduced echelon form if(i) all the zero rows come below the non-zero rows,(ii) the first non-zero entry in each non-zero row is 1 and occurs in a column to the right of the leading 1 in the row above,(iii) the leading 1 in each non-zero row is the only non-zero entry in the column that it is in.(If (i) and (ii) hold, the matrix is in echelon form.) For example, these two matrices are in reduced echelon form:Any matrix can be transformed to a matrix in reduced echelon form using elementary row operations, by a method known as Gauss–Jordan elimination. For any matrix, the reduced echelon form to which it can be transformed is unique. The solutions of a set of linear equations can be immediately obtained from the reduced echelon form to which the augmented matrix has been transformed. A set of linear equations is said to be in reduced echelon form if its augmented matrix is in reduced echelon form.

(i) all the zero rows come below the non-zero rows,

(ii) the first non-zero entry in each non-zero row is 1 and occurs in a column to the right of the leading 1 in the row above,

(iii) the leading 1 in each non-zero row is the only non-zero entry in the column that it is in.