(i) Let $A$ be an $n \times n$ complex matrix and $f(X)$ a polynomial with complex coefficients. By considering the Jordan normal form of $A$ or otherwise, show that if the eigenvalues of $A$ are $\lambda_{1}, \ldots, \lambda_{n}$ then the eigenvalues of $f(A)$ are $f\left(\lambda_{1}\right), \ldots, f\left(\lambda_{n}\right)$.

(ii) Let $B=\left(\begin{array}{llll}a & d & c & b \\ b & a & d & c \\ c & b & a & d \\ d & c & b & a\end{array}\right)$. Write $B$ as $B=f(A)$ for a polynomial $f$ with $A=\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right)$, and find the eigenvalues of $B$

[Hint: compute the powers of $A$.]