1. ## 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

2. ## 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.

3. ## 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.

4. ## Re: Prove or disprove

Recall that $\text{tr}(XY)=\text{tr}(YX)$.

5. ## 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?

6. ## Re: Prove or disprove

Originally Posted by Loser66
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.