# Give all the 2x2 row echelon reduced matrices

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## Homework Statement

Give all the possible 2x2 row echelon reduced matrices.

2. The attempt at a solution I thought about the matrices (0 0, 0 0), (1 0, 0 0), (0 1, 0 0), (0 0, 1 0), (1 0, 0 1), (0 1, 0 0), (0 0, 0 1). Where the "," inside the parenthesis means a change of row.
So in total I have found 7 matrices... is this right?
Hmm isn't there an infinity of them? Like for example (1 0, a, b) where a is different from 1.
As you see I'm confused.

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Hurkyl
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What is the definition of "row echelon reduced"?

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What is the definition of "row echelon reduced"?
From my notes : Suppose A is an mxn matrix. A is row echelon reduced if : 1) A is row reduced,
2) If s=#(1≤i≤m such that A(i,.)= the zero vector) ≥1 and r=m-s, then A(i,.)= zero vector for all i ≥r+1.
3)If r=#(1≤i≤m such that A(im.) different from the zero vector) ≥1, then min(1≤j≤n such that A(1,j) different from 0) < min(1≤j≤n such that A(2,j) different from 0) < ...< min(1≤j≤n such that A(r,j) different from 0).

What I understand is that 3) makes the echelon condition. And I don't understand 2)... Or maybe a little. Does it says that if there are more than one null column, they must be on the right of the matrix? I guess no...

HallsofIvy