Exercise 5.3.17

For the first Corollary, we may apply Theorem 5.18 to the matrix $A^t$ since we have the fact that $A$ and $A^t$ have the same characteristic polynomial. Thus we know that $\lambda =\nu (A)$. Also, the dimension of eigenspace of $A^t$ corresponding to $\lambda$ is $1$. But Exercise 5.2.13 tell us that $A^t$ and $A$ have the same dimension of the corresponding eigenspaces.

For the second Corollary, we know that $\nu (A)=1$. So if $\lambda \ne 1$ then we have $\left|\lambda \right|<1$ by Theorem 5.18 and its first Corollary. And the eigenspace corresponding $1$ has dimension one by the first Corollary.