A matrix is in reduced echelon form if it is in echelon form and, further, every column that contains a leading 1 (of some row) otherwise has entries of 0. For example, these two matrices are in reduced echelon form:

Any matrix can be transformed to a unique matrix in reduced echelon form using elementary row operations, by Gauss–Jordan elimination. The solutions of a system of linear equations can be immediately obtained from the reduced echelon form to which the augmented matrix has been transformed.