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Math Help - Maximazing determinant of symmetric matrix

  1. #1
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    Maximazing determinant of symmetric matrix

    Choose the number x to maximize the determinant (this means maximizing the
    entropy!) of the symmetric matrix
    S = 4 2 x
    2 4 2
    x 2 4

    How exactly are you supposed to do this without guessing?
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  2. #2
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    Re: Maximazing determinant of symmetric matrix

    Quote Originally Posted by kkar View Post
    Choose the number x to maximize the determinant (this means maximizing the
    entropy!) of the symmetric matrix
    S = 4 2 x
    2 4 2
    x 2 4

    How exactly are you supposed to do this without guessing?
    Do a cofactor expansion to get an expression for the determinant of S. This will be a polynomial in x. Maximise this function.
    Thanks from kkar and Ruun
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  3. #3
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    Re: Maximazing determinant of symmetric matrix

    Find the determinant of S. A trick in high school I learned gives me det(S) = [4*4*4 + 2*2*x + x*2*2] - [x*4*x + 2*2*4 + 4*2*2].

    Then det(S) = [64 + 4x + 4x] - [4x^2 + 16 + 16]
    Then det(S) = 64 + 8x - 4x^2 - 32
    Then det(S) = 32 + 8x - 4x^2

    Remember the maximum of ax^2 + bx + c = 0 where a < 0 occurs when x = -b/2a.

    The maximum det(S) occurs when x = -8/(2*-4) = -8/-8 = 1.

    Thus, x = 1.
    Thanks from kkar
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  4. #4
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    Re: Maximazing determinant of symmetric matrix

    That makes total sense. Thanks!
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