<s>[INST] How to compute eigenvalues of big $5×5$ matrix (symmetric matrix) . How to compute eigenvalues of big $5×5$ matrix (symmetric matrix) .
$\left(\begin{matrix} 
  0 & 0 & 0 & 0 & 1\\
  0 & 0 & 1 & 1 & 0\\
  0 & 1 & 0 & 1 & 0\\
  0 & 1 & 1 & 0 & 0\\
  1 & 0 & 0 & 0 & 0\\
\end{matrix}\right)
$
By characteristics equation will be lengthy process . If you are going to suggest "eigenvalues by inspection", please explain in steps .
 [/INST] There is a lot of $0$... you can compute it as usual...
$$\left|\begin{matrix}-t&0&0&0&1\\0&-t&1&1&0\\ 0&1&-t&1&0\\0&1&1&-t&0\\1&0&0&0&-t\end{matrix}\right|=
-t\left|\begin{matrix}
-t&1&1&0\\ 1&-t&1&0\\1&1&-t&0\\0&0&0&-t
\end{matrix}\right|+
\left|\begin{matrix}
0&0&0&1\\-t&1&1&0\\ 1&-t&1&0\\1&1&-t&0
\end{matrix}\right|$$
$$=-t^2\left|\begin{matrix}
-t&1&1\\1&-t&1\\1&1&-t
\end{matrix}\right|-
\left|\begin{matrix}
-t&1&1\\1&-t&1\\1&1&-t
\end{matrix}\right|
=-(t^2+1)\left|\begin{matrix}
-t&1&1\\1&-t&1\\1&1&-t
\end{matrix}\right|=...$$
</s>