Question 84
Let $P \in \mathbb{C}[X]$ be a polynomial with degree at least $2$ and $n \ge 2$ be an integer. Prove that, if $A \in \mathcal{M}_n(\mathbb{C})$ commutes with every matrix $B \in \mathcal{M}_n(\mathbb{C})$ having the property $P(B) = O_n$, then there exists $a \in \mathbb{C}$ such that $A = aI_n$.