# Prove or disprove

• May 1st 2014, 04:02 PM
Loser66
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
• May 2nd 2014, 12:24 PM
Loser66
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.
• May 2nd 2014, 12:56 PM
HallsofIvy
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.
• May 2nd 2014, 03:38 PM
emakarov
Re: Prove or disprove
Recall that $\text{tr}(XY)=\text{tr}(YX)$.
• May 3rd 2014, 07:41 AM
Loser66
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?
• May 3rd 2014, 08:44 AM
emakarov
Re: Prove or disprove
Quote:

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.