Exercise 2.7.21

Elimination on a symmetric 3 by 3 matrix leaves a symmetric lower right 2 by 2 matrix. $\left[\begin{array}{ccc}2\hfill & 4\hfill & 8\hfill \\ 4\hfill & 3\hfill & 9\hfill \\ 8\hfill & 9\hfill & 0\hfill \end{array}\right]$and $\left[\begin{array}{ccc}1\hfill & b\hfill & c\hfill \\ b\hfill & d\hfill & e\hfill \\ c\hfill & e\hfill & f\hfill \end{array}\right]$lead to $\left[\begin{array}{cc}\hfill -5\hfill & \hfill -7\hfill \\ \hfill -7\hfill & \hfill -32\hfill \end{array}\right]$ and $\left[\begin{array}{cc}d-b^2\hfill & e-\mathit{bc}\hfill \\ e-\mathit{bc}\hfill & f-c^2\hfill \end{array}\right]:$ symmetric!