Chapter 1.6, Problem 32E

Let $G$ be the $\left(2\times 2\right)$ matrix that follows, and consider any vector $\mathbf{x}$ in $R^2$ where both entries are not simultaneously zero:

$G=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right],\mathbf{x}=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right];\mathbf{ }\left|x_1\right|+\left|x_2\right|>0.$

Show that ${\mathbf{x}}^{T}G\mathbf{x}>0$. [Hint: Write ${\mathbf{x}}^{T}G\mathbf{x}$ as a sum of squares.]