### Row
Echelon Form

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

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

0 |
0 |
* |
# |
# |

* |
# |
# |
# |
# |

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

0 |
0 |
* |
# |
# |

0 |
* |
# |
# |
# |

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

0 |
* |
# |
0 |
0 |

* |
0 |
0 |
0 |
0 |

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

# |
# |
# |
* |
0 |

# |
# |
* |
0 |
0 |

Answer

Exercise Compute the (reduced) row echelon form and list the pivot columns.

[ |
0 |
0 |
0 |
1 |
] |

0 |
0 |
1 |
0 |

0 |
1 |
0 |
0 |

1 |
0 |
0 |
0 |

[ |
0 |
1 |
0 |
2 |
0 |
] |

0 |
0 |
1 |
0 |
2 |

0 |
0 |
0 |
1 |
0 |

0 |
0 |
0 |
0 |
0 |

[ |
1 |
0 |
0 |
0 |
] |

-1 |
1 |
0 |
0 |

0 |
-1 |
1 |
0 |

0 |
0 |
-1 |
1 |

[ |
1 |
2 |
1 |
1 |
] |

2 |
1 |
1 |
0 |

0 |
3 |
1 |
0 |

3 |
0 |
1 |
0 |

[ |
1 |
0 |
1 |
2 |
] |

1 |
4 |
-3 |
-2 |

1 |
-1 |
2 |
3 |

[ |
2 |
-3 |
6 |
2 |
5 |
] |

-2 |
3 |
-3 |
-3 |
-4 |

4 |
-6 |
9 |
5 |
9 |

-2 |
3 |
3 |
-4 |
1 |

[ |
1 |
-2 |
1 |
1 |
2 |
] |

-1 |
3 |
0 |
2 |
-2 |

0 |
1 |
1 |
3 |
4 |

1 |
2 |
5 |
13 |
5 |

[ |
1 |
-1 |
0 |
1 |
] |

-2 |
3 |
1 |
2 |

1 |
0 |
1 |
5 |

1 |
2 |
3 |
13 |

2 |
-2 |
4 |
5 |

[ |
1 |
2 |
3 |
0 |
1 |
1 |
] |

0 |
3 |
2 |
1 |
-1 |
0 |

-1 |
1 |
2 |
-2 |
1 |
0 |

1 |
-2 |
-2 |
1 |
0 |
0 |

4 |
1 |
3 |
2 |
2 |
1 |

Answer

Exercise Use different row operations to find
a (reduced) row echelon forms for the matrices in this example
and this example.
Compare your result with the ones obtained in the examples.

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

Exercise Check out the examples
of row echelon forms to convince yourself that

number of pivots = number of pivot rows = number of pivot columns.

Can you provide the reason in general?

Exercise** **Show that for a 3 by
3 matrix, all rows are pivot ⇔ all columns are pivot? What about general
`n` by `n` matrices?

Answer

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

Exercise How many shapes of `m` by `n`
row echelon forms are there?

Answer