Exercise 1.7.14

Let $S=\left[\begin{array}{ccc}\hfill a\hfill & \hfill d\hfill & \hfill e\hfill \\ \hfill d\hfill & \hfill b\hfill & \hfill f\hfill \\ \hfill e\hfill & \hfill f\hfill & \hfill c\hfill \end{array}\right]$, then we have

Solve for $a,b,c,d,e,f$, we have $S=\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -4\hfill & \hfill 8\hfill \\ \hfill -4\hfill & \hfill 4\hfill & \hfill -8\hfill \\ \hfill 8\hfill & \hfill -8\hfill & \hfill 16\hfill \end{array}\right]$

This matrix has pivots of $4,0,0$, it has rank of $1$, it’s determinant is zero. The eigenvalues are: $0,0,24$