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Why this property of the product of two matrices

  1. Feb 20, 2014 #1
    Hello,

    The product of a 2x5 matrix P and a 5x3 matrix B shall be a 2x3 zero matrix. P and B are all matrices of integers.

    P = [6 2 -5 -6 1;3 6 1 -6 -5]

    One possible B is [0 -4 0;3 0 0;-1 -1 3;2 -3 -2;1 1 3]

    This solution B has a property: det(PPt) = det(BtB) = 7778

    The question is: What does this property mean? How to get another B with this property?

    Senmeis
     
  2. jcsd
  3. Feb 20, 2014 #2

    HallsofIvy

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    I really don't know how you got B or why you refer to it as a "solution". Solution to what problem? Just having the property that det(PPt) = det(BtB)?
     
  4. Feb 20, 2014 #3
    Pb=0
     
  5. Feb 20, 2014 #4

    D H

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    Post-multiply B by any 3x3 matrix C, i.e., D=BC, and you'll have another 5x3 matrix D that satisfies PD=0. If C is unimodular (determinant = ±1), then det(D)=±det(B), so det(DTD)=det(BTB)=7778.

    See if you can take this further to ensure that all elements of D are integers.
     
    Last edited: Feb 20, 2014
  6. Feb 26, 2014 #5
    Yes, you are right, but I still don’t know if this property is by chance. More important, I can’t even get another B that fulfills this property.

    Senmeis
     
  7. Feb 26, 2014 #6

    D H

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    No, it is not by chance. Matrix multiplication is associative: (PB)C = P(BC). If PB=0 then P(BC) must necessarily be zero as well.
     
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