MathNet 2020 — Math Olympiad Vault — Part 05 Question 140

For $n \in \mathbb{N}$, $n \ge 2$, consider $A$ a matrix $n \times n$ with complex entries, such that $A^2 = \text{tr}(A) \cdot A$. Prove that the matrices $ABA$ and $ACA$ commute, for any $n \times n$ matrices $B$ and $C$, with complex entries.

Mihai Opincaru