## Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$

## Problem 559

For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$.

**(a)** $A=\begin{bmatrix}

4 & 2\\

2& 1

\end{bmatrix}$.

**(b)** $A=\begin{bmatrix}

2 & 1\\

1& 3

\end{bmatrix}$.