Exercise Do the pivot columns determine the shape of a row echelon form? Do the pivot rows determine the shape of a row echelon form?

Answer The pivot columns determine the shape of the row echelon form. For example, if we know the pivot columns of a 3 by 4 matrix are [col 1] and [col 3], then the shape must be

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

0 | 0 | * | # | ||

0 | 0 | 0 | 0 |

On the other hand, the pivot rows do not determine the shape of the row echelon form. For example, both of the following two shapes have [row 1] and [row 2] as the pivot rows.

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

0 | 0 | * | # | ||

0 | 0 | 0 | 0 |

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

0 | 0 | 0 | * | ||

0 | 0 | 0 | 0 |