A row of a matrix is called zero if all its entries are zero. Then a matrix is in echelon form if (i) all the zero rows come below the non‐zero rows, and (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. For example, these two matrices are in echelon form:

Any matrix can be transformed to a matrix in echelon form using elementary row operations, by a method known as Gaussian elimination. The solutions of a set of linear equations may be investigated by transforming the augmented matrix to echelon form. Further elementary row operations may be used to transform a matrix to reduced echelon form.