Angular momentum theory problem probably wrong sign

[lgnr]

I have to prove that \left\langle lm | \vec{\hat{r}} \times \vec{\hat{p}} | lm\right \rangle = \left\langle lm | \vec{\hat{p}} \times \vec{\hat{r}} | lm \right \rangle, where | lm \rangle are eigenkets of angular momentum operator \hat{L}^2

And I can't figure out a way to do this correctly. I wrote the angular momentum operator-vector in terms of its components, \hat{L_x}, \hat{L_y} and \hat{L_z}, and only the \hat{k} component survives the bra-ket operation, because I can write \vec{L_x} y \vec{L_y} in terms of ladder operators, and after lowering and rising the eigenstates, the corresponding eigenkets (except for the \hat{k} component) cancel with \langle lm | because of the orthogonality property of these eigenkets. Probably there is a wrong sign in the problem statement. Looks trivial in that case, but I want to know your opinion.

Thanks in advance for any advice.

 

[Bill_K]

Undoubtedly the wrong sign.