SEEMOUS 2022 Problem 3

Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$ , $A \neq O_n$ , be such that $$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$ where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$ , $|\alpha| \le 2$ . and $A\cdot A^* = A^*\cdot A.$