Is $\text{Tr}(U^{-1}) = \frac{1}{2} (\text{Tr}(U) ^2 - \text{Tr}(U^2))$ for $U \in SU(3) $?

Question

Is it true that $\text{Tr}(U^\dagger) = \frac{1}{2} (\text{Tr}(U) ^2 - \text{Tr}(U^2))$ for $U \in SU(3) $ ? In particular are there more general formulas/ systematic way of reducing higher powers of traces to expressions involving the complex conjugate of the trace ?

Answer 1

From the Cayley-Hamilton theorem, for any 3×3 matrix, not just a unimodular one, $$ A^3- (\operatorname{tr}A)A^2+\frac{1}{2}\left((\operatorname{tr}A)^2-\operatorname{tr}(A^2)\right)A-\det(A)I_3=O,\leadsto \\ A^{-1} \det(A)= A^2- (\operatorname{tr}A)A+\frac{1}{2}\left((\operatorname{tr}A)^2-\operatorname{tr}(A^2)\right) I_3. $$ so taking the trace for a unimodular matrix yields your formula.

You get the extension for generic n.