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Math Help - Determinant help

  1. #1
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    Determinant help

    Hello,

    I need help with this theory-based problem:

    Why is it not possible to find a square matrix such that:

    <br />
|AA^T| < 0

    I should say, from the properties I know of, |A| = |A^T|, what can I gather from this information?
    Last edited by Dranalion; January 15th 2010 at 09:24 PM.
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  2. #2
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    Quote Originally Posted by Dranalion View Post
    Hello,

    I need help with this theory-based problem:

    Why is it not possible to find a square matrix such that:

    <br />
det(AA^T) < 0

    (use properties of determinants to guide you)

    I should say, from the properties I know of, det(A) = det(A^T), what can I gather from this information?

    Any help is much appreciated,

    Dranalion
    Recall that |A| is a real scalar.

    You should know that |AB| = |A||B|.

    You should also know that |A^T| = |A|.


    So if you let B = A^T you have

    |AA^T| = |A||A^T|

     = |A||A|

     = |A|^2.


    Since |A| is a real number, and the square of any real number is nonnegative, this means that

    |A|^2 \geq 0 for all A.
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  3. #3
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    BTW I assume that the elements in A are all Real.

    Bringing in Complex elements may affect this result.
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  4. #4
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    Quote Originally Posted by Prove It View Post
    BTW I assume that the elements in A are all Real.

    Bringing in Complex elements may affect this result.
    Not if you replace A^T by A^*- that is, in addition to interchanging rows and columns, you also take the complex conjugate of each number. The "adjoint" of A for matrices over the real numbers is A^T. For matrices over the complex numbers, the adjoint is A^*.
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