Polynomial of $3\times 3$ matrix with $\operatorname{tr}(A)=\operatorname{tr}(A^2)=0$ and $\det(A)=1$ If $A$ be an invertible matrix $M_3(\mathbb{R})$ and we have $\operatorname{tr}(A)=\operatorname{tr}(A^2)=0$ and $\det(A)=1$, then what is the characteristic polynomial of $A$?

By Cayley-Hamilton, the characteristic formula is:
$$x^3 − \text{tr}(A)x^2+\frac{1}{2}[\text{tr}^2(A)−\text{tr}(A^2)]x−\det A =0$$
In particular, the characteristic polynomial becomes $x^3-1$.
