Exercise List all shapes of the row echelon forms
of 2 by 2 matrices ` A` so that

Answer We look for row echelon forms with no nonzero row (i.e., all rows pivot). For the 2 by 2 coefficient matrices, there is only one possibility.

[ | * | # | ] |

0 | * |

For the 2 by 3 coefficient matrices, we have the following possibilities.

[ | * | # | # | ] |

0 | * | # |

[ | * | # | # | ] |

0 | 0 | * |

[ | 0 | * | # | ] |

0 | 0 | * |

We cannot find 3 by 2 matrices with all rows pivot. (This means that any system of 3 equations in 2 variables is inconsistent for some right side.)

In general, for a system of equations to be always consistent, the number of equations cannot be more than the number of variables (see further discussion).