Exercise Explain why the following are not row echelon forms by showing how the shape can be improved.

1)

[ | 0 | * | # | # | # | ] |

0 | 0 | * | # | # | ||

* | # | # | # | # |

Answer The matrix is not a row echelon form because the rows are not arranged from the longest to the shortest. By [row 2] ↔ [row 3] and then [row 1] ↔ [row 2], we rearrange the rows in the right order and get a row echelon form.

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

0 | * | # | # | # | ||

0 | 0 | * | # | # |

2)

[ | 0 | * | # | # | # | ] |

0 | 0 | * | # | # | ||

0 | * | # | # | # |

Answer The matrix is not a row echelon form because
[col 2] can be further simplified. Specifically, let the two * numbers
in [col 2] to be `a` and `b`, then the operation (-`b`/`a`)[row 1] + [row 3]
gives us

[ | 0 | * | # | # | # | ] |

0 | 0 | * | # | # | ||

0 | 0 | ? | # | # |

which has simpler shape. Depending on whether ? is zero or not, the shape may or may not be a row echelon form. If ? is nonzero, further row operations are needed in order to get a row echelon form.

3)

[ | 0 | 0 | * | # | # | ] |

0 | * | # | 0 | 0 | ||

* | 0 | 0 | 0 | 0 |

Answer The matrix is not a row echelon form. Note that although [row 3] contains only one nonzero entry, its length, as counted from the leading entry, is 5. Similarly, the length of [row 1] and [row 2] are 3 and 4. Exchanging [row 1] and [row 3] gives us a row echelon form.

[ | * | 0 | 0 | 0 | 0 | ] |

0 | * | # | 0 | 0 | ||

0 | 0 | * | # | # |

4)

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

# | # | # | * | 0 | ||

# | # | * | 0 | 0 |

Answer The matrix (may or) may not be a row echelon form, again because the lengths of rows are counted from the leading entries. For example, if the [col 1] consists of 1, 1, 1,

[ | 1 | # | # | # | * | ] |

1 | # | # | * | 0 | ||

1 | # | * | 0 | 0 |

then all three rows have length 5. The lengths of [row 2] and [row 3] can be shortened (so that the matrix is simplified) by row operations such as -[row 1] + [row 2] and -[row 1] + [row 3].

[ | 1 | # | # | # | * | ] |

0 | # | # | # | # | ||

0 | # | # | # | # |

Remark The lesson we should learn from the third and the fourth problems is that zeros on the right of the matrix are irrelevant in counting the lengths of rows (and the determination of the row echelon form).