Chapter 2.6, Problem 2E

Show that the following symmetric matrices are not positive-definite by finding a vector $x\ne 0$ such that $x^{T}Ax<0$.

a. $\left[\begin{array}{cc}1& 0\\ 0& -3\end{array}\right]$

b. $\left[\begin{array}{cc}1& 2\\ 2& 2\end{array}\right]$

c. $\left[\begin{array}{cc}1& -1\\ -1& 0\end{array}\right]$

d. $\left[\begin{array}{ccc}1& 0& 0\\ 1& -2& 0\\ 0& 0& 3\end{array}\right]$