Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 1 / b / ii / Solution

Write $C=(c_{ij})$ and $(d_1,d_2,d_3)=(0,2,4)$. Because $D$ is a diagonal matrix,Thus $CD=DC$ exactly when $(d_j-d_i)c_{ij}=0$ for every $i,j$. The three eigenvalues are distinct, so every off-diagonal entry of $C$ must be zero; diagonal entries are unrestricted. Consequently the centralizer of a diagonal matrix here isEvery displayed matrix indeed commutes with $D$, since diagonal matrices multiply entry by entry on the diagonal.