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Row equivalence

  1. Apr 30, 2010 #1
    Every matrix is row equivalent to a unique matrix in echelon form?

    False, a matrix is row equivalent if it is non-singular.

    Is the above correct reasoning for the initial statement.
     
  2. jcsd
  3. Apr 30, 2010 #2

    Mark44

    Staff: Mentor

    If you start with any matrix and row reduce it to reduced row echelon form, is it possible to end up with two different matrices if you do different row operations? Notice that I've made a statement that's more restrictive than yours.
     
  4. Apr 30, 2010 #3
    I may not understand what you mean fully because if you reduce it echelon form it will come out the same no matter how we go about it.
     
  5. Apr 30, 2010 #4
    When you say it is unique do you mean there is a one-to-one relationship with every matrix to it's reduced echelon form or that there is only one reduced echelon form for any matrix?
     
  6. Apr 30, 2010 #5
    I don't know this was on one of my old test and I have my final on the 7th so I was just redoing everything and that is what it says word for word.
     
  7. Apr 30, 2010 #6

    Mark44

    Staff: Mentor

    Look up the definitions of echelon form and reduced row echelon form (or possible reduced echelon form). They are different.
     
  8. Apr 30, 2010 #7
    Depending on the book echelon form is upper triangular where the diagonals are 1 or depending on the book just upper triangular.

    rref is upper triangular with 1s in all the pivot rows and if possible diagonal entries only.
     
  9. Apr 30, 2010 #8
    Your reasoning is incorrect. Any identity matrix can be multiplied by a scalar to produce a singular, row equivalent matrix.
     
  10. Apr 30, 2010 #9

    Mark44

    Staff: Mentor

    The only scalar for which this is true is 0. There are only three row operations.
    1. Interchange row i and row j.
    2. Replace row i by a nonzero multiple of itself.
    3. Replace row i by itself plus k times row j.

    An identity matrix is never row equivalent to a singular matrix.
     
  11. May 1, 2010 #10
    So it's false because row operations can be carried out on a derived row equivalent row echelon matrix to produce another row equivalent row echelon matrix.
     
  12. May 1, 2010 #11

    Mark44

    Staff: Mentor

    Not necessarily. It depends on the definitions of echelon form and reduced echelon form he's using.
     
  13. May 1, 2010 #12
    That's what I can't get around with this problem. Would the context come from what he did in class?
     
  14. May 1, 2010 #13

    Mark44

    Staff: Mentor

    From the textbook he's using.
     
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