The eigenvalues can be found by solving the characteristic equation
$\begin{align*}\det \left[\left(\begin{array}{ccc}-\lambda & 1 & 0\\0& -\lambda & 1 \\1& 1/2 & -\lambd...$
This implies that answer (E) is true. The general solution to $\lambda^n = 1$ are $\exp\left(2 \pi i n/3 \right)$ where $n = 0, 1, 2, 3, \ldots, n-1$ . Therefore, the eigenvalues are
$\begin{align*}\lambda_1 &= 1 \\\lambda_2 &= \exp\left(2 \pi i /3 \right) \\\lambda_3 &= \exp\left(4 \pi i /3 \rig...$
$\lambda_2 \lambda_3 = 1$ , therefore, answer (C) is also true.
$\begin{align*}\lambda_1 \lambda_2 + \lambda_2 \lambda_3 + \lambda_1 \lambda_3 &= \exp\left(2 \pi i /3 \right) + 1 + \exp\...$
Therefore, answer (D) is also true.
$\begin{align*}\lambda_1 + \lambda_2 + \lambda_3 &= 1 + \exp\left(2 \pi i /3 \right) + \exp\left(4 \pi i /3 \right) \\&...$
Therefore, answer (A) is also true. Clearly, not all of the roots are real, so answer (C) is correct.

Solution to 1992 Problem 98