We use row operations to simplify matrices. A natural question is: How simple can a matrix become by row operations?

Example The following matrix

[ | 1 | 4 | 0 | 5 | ] |

0 | 0 | 2 | 6 | ||

0 | 0 | 0 | 3 |

has the upside down staircase shape,

[ | * | # | # | # | ] |

0 | 0 | * | # | ||

0 | 0 | 0 | * |

in which * denotes nonzero numbers and # denotes any numbers. We claim that the shape cannot be improved by row operations.

First, the row operations `r`[row 1] + [row 2], `r`[row 1] + [row 3], `r`[row 2] + [row 3], with `r`
≠ 0, produce the following shapes.

[ | * | # | # | # | ] |

* | # | # | # | ||

0 | 0 | 0 | * |

[ | * | # | # | # | ] |

0 | 0 | * | # | ||

* | # | # | # |

[ | * | # | # | # | ] |

0 | 0 | * | # | ||

0 | 0 | * | # |

All are more complicated than the original one. Moreover, adding multiples of lower rows to upper rows, such as `r`[row 3] + [row 1],
will give us the same shape.

Next, the operation [row 2] ↔ [row 3] gives us

[ | * | # | # | # | ] |

0 | 0 | 0 | * | ||

0 | 0 | * | # |

Instead of simplifying the shape, this makes the shape uglier. You may check out the effects of the other exchange operations.

Finally, multiplying a nonzero number to a row does not change the shape, so that the shape is still not improved.

The following summarizes the simplest *shape* one can get by row operations.

A row echelon form is characterized by two properties

- The shape cannot be further improved by row operations
- The rows are arranged from the "longest" to the "shortest"