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Math Help - Symettric Matrices

  1. #1
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    Symettric Matrices

    I was going over some practice problems in my book and I got stuck on this particular one.

    It says:

    Let A and B be n * n symmetric matrices. Show that AB is symmetric if and only if A and B commute.

    Any help would be appreciated!
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  2. #2
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    Let me proof it one way, that should show you how do the other direction.

    I shall use "T" to represent the transpose.

    By hypothesis A and B are symmetric.
    Thus, T(A)=A and T(B)=B by definition.

    We show that if AB is symmetric then AB commute.
    That is T(AB)=AB
    But by a theorem we know that,
    T(AB)=T(B)T(A)
    Thus,
    T(B)T(A)=AB
    But T(B)=B and T(A)=A because they are symmetric.
    Thus,
    BA=AB
    And hence they commute.
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