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Thread: 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:

    $\displaystyle
    |AA^T| < 0$

    I should say, from the properties I know of, $\displaystyle |A| = |A^T|$, what can I gather from this information?
    Last edited by Dranalion; Jan 15th 2010 at 08:24 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:

    $\displaystyle
    det(AA^T) < 0$

    (use properties of determinants to guide you)

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

    Any help is much appreciated,

    Dranalion
    Recall that $\displaystyle |A|$ is a real scalar.

    You should know that $\displaystyle |AB| = |A||B|$.

    You should also know that $\displaystyle |A^T| = |A|$.


    So if you let $\displaystyle B = A^T$ you have

    $\displaystyle |AA^T| = |A||A^T|$

    $\displaystyle = |A||A|$

    $\displaystyle = |A|^2$.


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

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

    Bringing in Complex elements may affect this result.
    Not if you replace $\displaystyle A^T$ by $\displaystyle A^*$- 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 $\displaystyle A^T$. For matrices over the complex numbers, the adjoint is $\displaystyle A^*$.
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