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Math Help - Prove Matrices are symmetric

  1. #1
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    Prove Matrices are symmetric

    I don't know how to prove this if and only if statement.

    Let A & B be symmetric matrices.
    Show AB is symmetric if and only if AB=BA.

    Any help is much appreciated!
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  2. #2
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    You could remember that the product of two symmetric matrices are symmetric iff the matrices commute.

    (AB)^{t}=B^{t}A^{t}=BA
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  3. #3
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    I got that...but I don't know how to express this.

    Ok, I know in an if and only if proposition, then you have to prove the statement both ways (forwards and backwards).

    So....Assume AB is symmetric. Show AB=BA
    (AB)^T=(B^T)(A^T). We know A & B are symmetric so,
    AB=BA

    Then you have to go the other way...
    Assume AB=BA, show AB is symmetric.
    So do I do the same this basically as before to show this?
    (AB)^T=(B^T)(A^T)=BA

    Thanks.
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