This question is from an old comprehensive exam
Let $A$ be an $n\times n$ complex matrix, which commutes with all diagonalizable $n\times n$ complex matrices. Prove that $A$ is a constant times the identity matrix. Hint: What matrices commute with all diagonal matrices?
The hint says to find the matrices that commute with all diagonal matrices, which should themselves be diagonal.
What I've tried: Let $A \in M_n(\mathbb{C})$, $B$ be a diagonalizable $n\times n$ complex matrix such that $P^{-1}BP = D$, a diagonal matrix.
As $AB =BA$ then $P^{-1}ABP =P^{-1}BAP$, so $$P^{-1}APP^{-1}BP =P^{-1}BPP^{-1}AP$$ Which means that $$(P^{-1}AP)D = D(P^{-1}AP)$$ And so $P^{-1}AP$ is diagonal, so $A$ is diagonalizable, and moreover is simultaneously diagonalizable.
But from here I'm stuck. I know that $A$ commutes with all diagonalizable matrices, so this should always work, but I don't see how to get that $A$ is a scalar matrix.
Am I missing something obvious? Thank you in advance for your help.