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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- May 1st 2014, 05:02 PMLoser66Prove 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 - May 2nd 2014, 01:24 PMLoser66Re: 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. - May 2nd 2014, 01:56 PMHallsofIvyRe: 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.

- May 2nd 2014, 04:38 PMemakarovRe: Prove or disprove
Recall that $\text{tr}(XY)=\text{tr}(YX)$.

- May 3rd 2014, 08:41 AMLoser66Re: 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? - May 3rd 2014, 09:44 AMemakarovRe: Prove or disprove