Exercise List all shapes of the row echelon forms
the the augmented matrices of *consistent* systems of 2 equations and
2 variables. What about systems of other sizes?

Answer For 2 equations in 2 variables, we look for all 2 by 3 matrices with no rows of the form [0 0 ≠0]. The following are all such shapes.

[ | * | # | # | ] |

0 | * | # |

[ | * | # | # | ] |

0 | 0 | 0 |

[ | 0 | * | # | ] |

0 | 0 | 0 |

[ | 0 | 0 | 0 | ] |

0 | 0 | 0 |