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Linear Algebra: Determinant

  1. Feb 21, 2012 #1
    I need help with another homework problem

    Let n be a positive integer and An*n a matrix such that det(A+B)=det(B) for all Bn*n. Show that A=0

    Hint: prove property continues to hold if A is modified by any finite number of row or column elementary operations

    It seems obvious that A=0 but i'm having trouble developing the proof. Any help would be great.
     
  2. jcsd
  3. Feb 21, 2012 #2

    micromass

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    Please post homework in the homework forum. I moved it for you now.

    A hint for the proof: can you write a row/column operation as an elementary matrix??
     
  4. Feb 21, 2012 #3
    Yes and the product of the elementary matrices returns
    A=E1*E2*..*En

    is this what you are referring to?
     
  5. Feb 21, 2012 #4

    micromass

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    Yes. Let E be an elementary matrix, can you show that

    [tex]det(EA+B)=det(B)[/tex]

    ??
     
  6. Feb 21, 2012 #5
    Im not quite sure how to show this
     
  7. Feb 21, 2012 #6

    micromass

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    Hint: [itex]B=EE^{-1}B[/itex].

    Use that [itex]det(XY)=det(X)det(Y)[/itex].
     
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