Let $A$ be a diagonal matrix whose characteristic polynomial is $$P(x) = (x − 16)^8(x − 15)^8(x − 14)^7(x − 13)^7(x − 12)^6(x − 11)^6· · ·(x − 2)(x−1)$$ Let $V$ be the set of all $72\times 72$ matrices commuting with $A$. Let $W$ be the subset of $V$ consisting of all diagonal matrices.
$(a)$ find the dimension of $W ?$
$(b)$ find the dimension of $V $
My attempt : For $a)$: Dimension of $$W = 2 \sum_{k=1}^{8} k^2= 2.\frac{8(8+1)(16+1)}{6}=408$$
For $b)$ : Dimension of $V = 72$
Is it correct ?
Any hints/solution will be appreciated
Thank you!