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Matrices in reduced row-echelon form

  1. 1. The problem statement, all variables and given/known data
    Determine which of the matrices below are in reduced row-echelon form:
    a)
    1_2_0_2_0
    0_0_1_3_0
    0_0_1_4_0
    0_0_0_0_1

    b)
    0_1_2_0_3
    0_0_0_1_4
    0_0_0_0_0

    c)
    1_2_0_3
    0_0_0_0
    0_0_1_2

    d)
    0_1_2_3_4



    2. Relevant equations



    3. The attempt at a solution
    Okay, so I know for sure that (a) is not in reduced row-echelon form because the leading one in row 2 had a nonzero value in its column.

    (b) is in reduced row-echelon form.

    (c) is the one I'm most curious about. I feel that since there is a leading 1 in row 3, but that there is no leading 1 in row two (and to the left at that) that it's not in reduced row-echelon form, is this correct?

    (d) I feel confident is in reduced row-echelon form.

    Are all my thoughts correct?
     
  2. jcsd
  3. radou

    radou 3,217
    Homework Helper

    If it is so, what do you propose to reduce it? :wink:
     
  4. To reduce it would you switch row 2 and 3?
     
  5. radou

    radou 3,217
    Homework Helper

    You do not need to switch row 2 and 3. Only for the sake of aesthetics, perhaps. The system if completely reduced and you can read out all relevant information from the matrix.
     
  6. radou is right: you do not need to swap row 2 and row 3, but "reduced row echelon form" has a formal definition and it is necessary to swap row 2 with row 3 to put c) in that form.
     
  7. HallsofIvy

    HallsofIvy 40,425
    Staff Emeritus
    Science Advisor

    Well, I agree with the last part of what jalexanal said but, as a result, I would have to say "radou is wrong"! Just for solving equations or related problems, the order of rows does not matter but for this problem, to tell whether or not the matrices are in "reduced row echelon" form, it does matter. In reduced row echlon form, there must be no rows below a given row with first non-zero member further to the left.
     
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