Let $A$ be a $55 \times 55$ diagonal matrix with characteristic polynomial $(x - c_1)(x - c_2)^2(x - c_3)^3\cdots(x - c_{10})^{10} $ where $c_1,\cdots,c_{10}$ are all distinct. Let $V$ be the vector space of all $55 \times 55$ matrices $B$ such that $AB = BA$. What is the dimension of $V$ ?
Try:
Since $A$ is given to be diagonal matrix and $AB = BA \implies A$ and $B$ share same eigenbasis, hence are simultaneously diagonalizable. $\exists\, P$ non singular such that $P^{-1}AP$ and $P^{-1}BP$ are diagonal. Using the map $f:V \rightarrow A$ as $f(B)=P^{-1}BP \quad \forall B\in V$, we get dimension of $V$ as 10.