Find $A$ when $A^2$ is a $2\times 2$ zero-matrix and $A$ is symmetric. 
$A^2 = 
       \begin{pmatrix}
        0 & 0 \\
        0 & 0 
        \end{pmatrix}$ and $A$ is symmetric. Find the matrix $A$.

Well, I know that $A = A^T$ and $A^2 = 
       \begin{pmatrix}
        0 & 0 \\
        0 & 0 
        \end{pmatrix}
$. I'm not sure how to continue from here. Any suggestions?
Btw, I thought of something: $A^2 = AA = A A^T = I$ but it is wrong.

Here's a neat approach: for any matrix $A$ with entries $a_{ij}$, 
$$
\text{trace}(A^TA) = \sum_{i,j=1}^n a_{ij}^2
$$
In our case, we know that $A^TA = 0$.
