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Linear Algebra scholarship exam

  1. Dec 7, 2011 #1
    1. The problem statement, all variables and given/known data

    This is one of the three linear algebra scholarship questions given by my university last year. I've solved the other two, but this one is posing a bit of a problem. Question 1 on the file.


    Given that for a nxn matrix A both matricies A and A-1 have integer entries, show that det(A) = +-1

    2. Relevant equations



    3. The attempt at a solution

    I'm completely lost. I have a feeling co-factor expansion isn't the way to go, as that would be very messy, and the other two worked out fairly nicely when you know what you're doing.
     

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  3. Dec 7, 2011 #2

    micromass

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    Can you show that det(A) and det(A-1) are integers??
     
  4. Dec 7, 2011 #3
    I can reason it now, but not really mathematically show it.
    All the entries of A are integers. So by cofactor expansion for det(A) along the first row every part will be a product of integers, so an integer. Same reasoning for det(A-1)

    But det(A-1) = det(A)-1

    So, an integer = 1/that integer, so det(A) =1
     
  5. Dec 7, 2011 #4

    micromass

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    or -1.

    That is indeed the correct reasoning!!
     
  6. Dec 7, 2011 #5
    I can't edit for some reason, but I meant +-1
     
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