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Math Help - Prove or disprove

  1. #1
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    Prove or disprove

    I need help on this problem:
    Prove or find an example to the contrary. There are no complex 3x3 matrices X and Y such that
    XY-YX= \(\left[\begin{matrix}1&0&0\\0&-2&0\\0&0&3\end{matrix}\right]\)
    I don't know how to start. Please, help
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  2. #2
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    Re: Prove or disprove

    I got it anyway. Ha, it's quite easy. I was scared.
    It is just 1 line proof.
    Det (XY) = Det X * Det Y , so that the left hand side =0 when the right hand side is not. So that, there are no any (Real or Complex) matrices satisfy the condition.
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  3. #3
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    Re: Prove or disprove

    It is true that det(XY)= det(X)det(Y). But it is NOT generally true that det(A- B)= det(A)- det(B) so your proof does not hold. XY- YX does NOT necessarily have determinant 0.
    Thanks from topsquark and Loser66
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  4. #4
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    Re: Prove or disprove

    Recall that $\text{tr}(XY)=\text{tr}(YX)$.
    Thanks from romsek, johng, topsquark and 1 others
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  5. #5
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    Re: Prove or disprove

    @HallsofIvy oh, yea, you are right let me think more
    @emakarov : Thanks a lot
    My new logic :
    tr(XY ) = tr(YX)--> tr (XY) - tr(YX) =tr(XY-YX) =0 while tr(right hand side matrix ) \(\neq\)0
    Does it work?
    Last edited by Loser66; May 3rd 2014 at 08:10 AM.
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  6. #6
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    Re: Prove or disprove

    Quote Originally Posted by Loser66 View Post
    tr(XY ) = tr(YX)--> tr (XY) - tr(YX) =tr(XY-YX) =0 while tr(right hand side matrix ) \(\neq\)0
    Does it work?
    Yes, this is correct.
    Thanks from Loser66
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