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Math Help - Invertibility of AB+BA

  1. #1
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    Invertibility of AB+BA

    I have to show that AB+BA is invertible if A and B are invertible. I tried to work with determinants but I never manage to get rid of that nasty plus . Maybe someone has some suggestions? Thanks!
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  2. #2
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    Re: Invertibility of AB+BA

    What you're attempting to prove isn't true.

    \text{Let } A = \left(\begin{matrix} 1& 2 \\ -2& -1 \end{matrix}\right), B = \left(\begin{matrix} 0& 1 \\ 1& 0 \end{matrix}\right).

    \text{Then } det(A) = (1)(-1) - (-2)(2) = 3, \ det(B) = (0)(0) - (1)(1) = -1,

    \text{ so both } A \text{ and } B \text{ are invertible in } M_{2, 2}(\mathbb{R}).

    \text{Have } AB = \left(\begin{matrix} 1& 2 \\ -2& -1 \end{matrix}\right) \left(\begin{matrix} 0& 1 \\ 1& 0 \end{matrix}\right) = \left(\begin{matrix} 2& 1 \\ -1& -2 \end{matrix}\right)

    \text{and } BA = \left(\begin{matrix} 0& 1 \\ 1& 0 \end{matrix}\right) \left(\begin{matrix} 1& 2 \\ -2& -1 \end{matrix}\right) = \left(\begin{matrix} -2& -1 \\ 1& 2 \end{matrix}\right),

    \text{so that } AB + BA = \left(\begin{matrix} 0& 0 \\ 0& 0 \end{matrix}\right), \text{ which is clearly not invertible.}
    Last edited by johnsomeone; October 16th 2012 at 02:29 PM.
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