Exercise 8.C.6

Proof. Define $T\in L({ℂ}^4)$ by

Then, the standard basis of ${ℂ}^4$ consists of eigenvectors of $T$ corresponding to the eigenvalues $0,1,1,3$. Applying $T(T-I)(T-3)$ to each of these basis vectors shows that $T(T-I)(T-3)=0$. Hence $z(z-1)(z-3)$ is the minimal polynomial of $T$. We have

Thus, by Exercise 11 in section 8B, the characteristic polynomial of $T$ is $z{(z-1)}^2(z-3)$. □