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Linear Algebra and rank problem.

  1. Apr 21, 2006 #1
    I have the following problem which I can't figure out.

    Let A = [a_11,a_12;a_13; a_21; a_22; a_23;]
    Show that A has rank 2 if and only if one or more of the determinants

    | a_11,_a_12; a_21,a_22| , |a_11,a_13;a_21,a_23|,|a_12,a_13;a_22,a_23|
    I know its a 2x3 matrix..which the det. wouldn't apply since it is not square. Not sure how to proceede
     
  2. jcsd
  3. Apr 21, 2006 #2

    HallsofIvy

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    You might start by stating the problem correctly

    "Show that A has rank 2 if and only if one or more of the determinants
    | a_11,_a_12; a_21,a_22| , |a_11,a_13;a_21,a_23|,|a_12,a_13;a_22,a_23|" is what?? What is supposed to be true about them?

    "I know its a 2x3 matrix..which the det. wouldn't apply since it is not square."
    That's irrelevant- the problem doesn't say anything about the determinant of A (which doesn't exist) only the determinants of those 2 by 2 subsets.
     
  4. Apr 21, 2006 #3
    Yeah i states how they had it though. sorry.
    they said.. the det != 0
     
  5. Apr 22, 2006 #4

    HallsofIvy

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    Okay, now that we have that straightened out, exactly what is your definition of "rank"? What do you get if you "row reduce" A?
     
  6. Apr 22, 2006 #5
    the rank is the dimensions of the row space and column space of a matrix.
    So when we do r-r-e, wherever we get leading ones, that is the rank #.
     
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