A rectangular matrix is in **echelon
form** if

- All nonzero rows are above any rows of all zeros.
- Each leading entry of a row is in a column to the right of the leading entry of the row above it.
- All entries in a column below a leading entry are zero.

The matrix is in **reduced echelon
form** if it
satisfies the above requirements and also

- The leading entry in each row is one.
- The entries above each leading entry are zero as well.

Examples (x represents any nonzero entry, while * represents any entry):

x | * | * | * | * |

0 | 0 | 0 | x | * |

0 | 0 | 0 | 0 | 0 |

is in echelon form, while

1 | * | * | 0 | * |

0 | 0 | 0 | 1 | * |

0 | 0 | 0 | 0 | 0 |

is in reduced echelon form.