P3404
We have
\begin{equation}
\text{det}(A - \lambda I_2)
=
\text{det}
\begin{bmatrix}
- \lambda & - 1 \\
1 & - \lambda
\end{bmatrix}
= \lambda^2 + 1
\end{equation}
The equation $\lambda^2 + 1 = 0$ has no real solutions, but we have
\begin{equation}
i^2 + 1
= -1 + 1
= 0
\end{equation}
and
\begin{equation}
(-i)^2 + 1
= i^2 + 1
= 0
\end{equation}
$\square$
