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Give all the 2x2 row echelon reduced matrices

  1. Jan 18, 2009 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    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.
     
  2. jcsd
  3. Jan 18, 2009 #2

    Hurkyl

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    What is the definition of "row echelon reduced"?
     
  4. Jan 18, 2009 #3

    fluidistic

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    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...
     
  5. Jan 19, 2009 #4

    HallsofIvy

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    What that tells you is that (0 0, 1 0) , (0 0, 0 1), and (0 1, 1, 0) are not "row-reduced echelon matrices", the first two by (2) and the third by (3).
     
  6. Jan 19, 2009 #5

    fluidistic

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    Thank you. I also checked out wikipedia's definition of what is a row-reduced echelon matrix and I found it more clearer than my notes. So thanks to both.
     
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