Exercise 1.7.17

To have a negative eigenvalue, it means the matrix is not positive definite, so for matrix $S=\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill b\hfill & \hfill c\hfill \end{array}\right]$, we need $ac-b^2>0$, so $b>{(ac)}^{\frac{1}{2}}$, on the other hand, we need $b<\frac{a+c}{2}$. Pick $a=1$ and $c=4$, we see that $b=2.4$ satisfies the requirement.

Solve for eigenvalues we have $