I have been trying to solve the following problem.
Let $\{v_{1},v_{2},....,v_{16}\}$ be an ordered basis for $V=\mathbb{C}^{16}$.If $T$ is a linear transformation on $V$ defined by $T(v_{i})=v_{i+1}$ for $1\leq i\leq 15$ and $T(v_{16})=-(v_{1}+v_{2}+....+v_{16}).$
Then which of the following is/are true?
(a)$T$ is singular with rational eigenvalues,
(b)$T$ is singular but has no rational eigenvalues,
(c)$T$ is regular(invertible) with rational eigenvalues,
(d)$T$ is regular but has no rational eigenvalues.
Could someone point me in the right direction(e.g. a certain theorem or property I have to use?) Any kind of hints will be helpful.