Non singular matrices with trace property

Question

If $A,B\in M_{3}\Bbb{C}$ are non singular matrices such that $$A^2-\mbox{ tr }A(A)+A^*=B^2-\mbox{ tr }B(B)+B^*,$$ show that $$A^*=B^*$$ we know that $$A^3-(\mbox{ tr }A)A^2+(\mbox{ tr adj}A)A+detA.I_3=0$$

Answer 1

This is not true. Let $a=\frac12(1+\sqrt{5})$ and $b=\frac12(1-\sqrt{5})$. Then the equation $$ X^2-\operatorname{tr}(X)X+X^\ast=\operatorname{diag}(1,1,0) $$ has two non-singular solutions $A=\operatorname{diag}(a,b,1)$ and $B=\operatorname{diag}(b,a,1)$.