<s>[INST] Eigenvalue of (some) $ 4 \times 4 $ symmetric matrices $$A=\pmatrix{
0  & 3 & 2 &  0  \\
3  & 0 &  0 & 2  \\
2  & 0 &  0 & 3  \\
0  & 2 & 3 &  0  \\
}$$
Is there a quicker way to compute eigenvalues of this matrix other than to do it the long way? And what are the strategies for similar matrices?
 [/INST] Observe
\begin{align}
M=
\begin{pmatrix}
A & B\\
B & A
\end{pmatrix}
\end{align}
where $B = 2I_2$ and $A=\begin{pmatrix}0 & 3\\ 3 & 0 \end{pmatrix}$ and $A$ & $B$ commute. Then we see that
\begin{align}
\det\left(M-\lambda I_4 \right) = \det ((A-\lambda I_2)^2-B^2)
\end{align}
where we used the determinant formula for block matrices.
Note that
\begin{align}
(A-\lambda I_2)^2=
\begin{pmatrix}
\lambda^2+9 & -6\lambda\\
-6\lambda & \lambda^2+9
\end{pmatrix}
\end{align} 
which means
\begin{align}
\det (M-\lambda I_4) = (\lambda^2+5)^2-36\lambda^2 = (\lambda^2-1^2).(\lambda^2-5^2)
\end{align}
</s>