What is the numerical range of $A$? Let $A = \left( {\begin{array}{*{20}{c}}
   0 & 0 & 0 & 1  \\
   0 & 0 & 1 & 0  \\
   0 & 1 & 0 & 0  \\
   1 & 0 & 0 & 0  \\
\end{array}} \right)$
What is the  numerical range of $A$?

Hint: This matrix is symmetric $A^T=A$, so in particular $$A^TA=AA^T$$ which means that the matrix is normal (as real valued). So, its numerical range is the convex hull of its eigenvalues, see here bullet 10. Now, $$\det(A-λΙ)=(λ^2-1)^2$$
