A matrix is in row echelon form if it has the following characteristics:
1. All zero rows are at the bottom of the matrix.
2. The leading entry of each nonzero row after the first occurs to the right of the leading entry of the previous row.
3. All entries below a leading entry are zero.
Some conventions additionally require each leading entry to be 1. A matrix is in reduced row-echelon form if it is in row echelon form, every leading entry is 1, and every leading 1 is the only nonzero entry in its column. Gaussian elimination produces a row-echelon form, while Gauss-Jordan elimination continues the reduction to reduced row-echelon form. The row-echelon form of a matrix is generally not unique, but its reduced row-echelon form is unique.
The positions of the pivot elements in the reduced row-echelon form identify the pivot variables, while columns without pivots identify the free variables. When the pivot-column indices are applied to the original matrix, the corresponding original columns form a basis for its column space (Piziak and Odell 1999).
