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Math Help - Prove there exists a matrix

  1. #1
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    Prove there exists a matrix

    Hi. Here is a problem I found in my algebra book and I don't know how to solve it. Could you please help me?

    Show that there exists a matrix A \in M(n,n;R), such that m_{ij} \in \{-1,0,1\} and det M=1995

    My problem is that I don't know what I should do to prove that there exist a certain matrix.
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  2. #2
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    Re: Prove there exists a matrix

    I may be oversimplifying things, but I think that all you need to do is show a matrix for which this is the case.
    You need to find a square matrix (obviously), in which all entries that aren't on the diagonal are either -1, 0 or 1.
    This means you could just give a diagonal or triangular matrix, with values on the diagonal that when multiplied together, amount to 1995.
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  3. #3
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    Re: Prove there exists a matrix

    Quote Originally Posted by Zangeki View Post
    You need to find a square matrix (obviously), in which all entries that aren't on the diagonal are either -1, 0 or 1.
    Why did you decide that the diagonal elements can be something other than -1, 0 or 1?

    Quote Originally Posted by Zangeki View Post
    This means you could just give a diagonal or triangular matrix, with values on the diagonal that when multiplied together, amount to 1995.
    Well, if you multiply -1, 0 and 1, you won't get 1995...

    Edit: Added the first remark.
    Last edited by emakarov; January 10th 2013 at 11:24 AM.
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  4. #4
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    Re: Prove there exists a matrix

    Yes, you are over simplifying. The determinant of a diagonal or triangular matrix is just the product of the numbers on the diagonal.
    If the numbers in a diagonal or triangular matrix are only -1, 0, or 1, the only possible determinants are -1, 0, and 1 so that is not possible.
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  5. #5
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    Re: Prove there exists a matrix

    Ah, I assumed i couldn't be equal to j. My bad.
    Last edited by Zangeki; January 10th 2013 at 11:37 AM. Reason: bold
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  6. #6
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    Re: Prove there exists a matrix

    I'm sorry. I made a mistake above but I cannot edit the post. There should be detA=1995, not detM
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  7. #7
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    Re: Prove there exists a matrix

    I have replied to this problem in another forum.
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