MathNet 2010 — Part 05 Question 25

Let $A \in \mathcal{M}_3(\mathbb{C})$ be such that $\text{tr}(A^2) = \text{tr}(A^*)$. Show that there exist $\alpha, \beta \in \mathbb{C}$ such that the matrix $(A + \alpha I_3)^3 + \beta I_3$ is nilpotent.