What are all of the reduced row echelon forms of a 3x3 matrix?

Question

I have some, but I don't want to miss any!

For example, a reduced row echelon form of a 3x3 matrix could be

1 0 0
0 1 0
0 0 1
or
1 0 7
0 1 0
0 0 0

etc.

Answer 1

Rank 3:

The 3x3 identity matrix.
--------------------------------
Rank 2:

Any matrix of the form

(1 0 *)
(0 1 *)
(0 0 0)

The matrix

(0 1 0)
(0 0 1)
(0 0 0).
-------------------------------------
Rank 1:

(1 * *)
(0 0 0)
(0 0 0)

(0 1 *)
(0 0 0)
(0 0 0)

(0 0 1)
(0 0 0)
(0 0 0)
----------------------------------------
Rank 0:

The 3x3 zero matrix.

Answer 2

You may want to be more specific on this one. If you give an example of a matrix we can discuss all of them. The thing is row reduced echelon forms are not unique as there are many ways to row reduce a matrix. On the other hand I can't think of a reason why anyone would care about having all of them. Having just one would help any question you might want to answer about a matrix.

Answer 3

Echelon Forms