Let $A$ be a complex $n \times n$ matrix with an eigenvalue $\lambda$. Show directly from the definitions that:

(i) $A^{r}$ has an eigenvalue $\lambda^{r}$ for any integer $r \geqslant 1$; and

(ii) if $A$ is invertible then $\lambda \neq 0$ and $A^{-1}$ has an eigenvalue $\lambda^{-1}$.

For any complex $n \times n$ matrix $A$, let $\chi_{A}(t)=\operatorname{det}(A-t I)$. Using standard properties of determinants, show that:

(iii) $\chi_{A^{2}}\left(t^{2}\right)=\chi_{A}(t) \chi_{A}(-t)$; and

(iv) if $A$ is invertible,

$$\chi_{A^{-1}}(t)=(\operatorname{det} A)^{-1}(-1)^{n} t^{n} \chi_{A}\left(t^{-1}\right)$$

Explain, including justifications, the relationship between the eigenvalues of $A$ and the polynomial $\chi_{A}(t)$.

If $A^{4}$ has an eigenvalue $\mu$, does it follow that $A$ has an eigenvalue $\lambda$ with $\lambda^{4}=\mu$ ? Give a proof or counterexample.