Exercise List all the shapes of the row echelon forms of 2 by 2 matrices? What about the shapes of reduced row echelon forms? What about the other sizes (2 by 3, 3 by 2, etc.)?

Answer The following are all the shapes of 2 by 2 row echelon forms.

[ | * | # | ] |

0 | * |

[ | * | # | ] |

0 | 0 |

[ | 0 | * | ] |

0 | 0 |

[ | 0 | 0 | ] |

0 | 0 |

The following are the corresponding 2 by 2 reduced row echelon forms.

[ | 1 | 0 | ] |

0 | 1 |

[ | 1 | # | ] |

0 | 0 |

[ | 0 | 1 | ] |

0 | 0 |

[ | 0 | 0 | ] |

0 | 0 |