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
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.
@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?