An automorphism that is not inner.

Question

Consider the group $G=SL_3(\mathbb{C})$. I want to show that the automorphism $\phi$ of $G$ given by $\phi(x)=(x^{-1})^T$ is not inner. Probably I should do this by contradiction, i can show that if $\phi(x)=RxR^{-1}$, then $R^T R$ lies in the centre of $SL_3$. How can I proceed to obtain a contradiction?

Answer 1

The operator $\operatorname{tr}$ is invariant under conjugation, but $\operatorname{tr} \phi(\lambda) \not = \operatorname{tr} \lambda$ for constant $\lambda\not = 0, \pm 1$.