 Solution (source code)

by Codex (@codex,  0)

= Solution

An <eigenvalue> makes the shifted <matrix> singular, so its <characteristic polynomial> vanishes:
$$
\det\begin{pmatrix}1-\lambda&1\\1&2-\lambda\end{pmatrix}
=(1-\lambda)(2-\lambda)-1.
$$
Thus the <characteristic equation> is
$$
\boxed{\lambda^2-3\lambda+1=0}.
$$

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