Find matrix with given properties

Question

I proved that for a positive semidefinite matrix holds: $$ x^TAx =0 \Rightarrow Ax=0. $$

I wanted to come up with an counterexample for an indefinite say $2 \times 2$ matrix:

So there is a $x$, such that $$ x^TAx =0, $$ but $Ax\ne 0$.

How can I construct such a counterexample.

Answer 1

Let $$A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$$

And let $x=(0,1)$

Answer 2

Take $A=\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}$ and the vector $x=\begin{pmatrix}1 & 0 \end{pmatrix}$