linear algebra, hard for me

If $A = \left( {a_{ij} } \right)$ and $A^2 = \left( {b_{ij} } \right),\,\,b_{ij} = \sum\limits_{k = 1}^n {\left( {a_{ik} } \right)\left( {a_{kj} } \right)}$ .
Because this is a symmetric matrix $a_{ik} = a_{ki}$ if the diagonal is zero consider the following.
$\left( {\forall i} \right)\;0 = \left( {b_{ii} } \right) = \sum\limits_{k = 1}^n {\left( {a_{ik} } \right)\left( {a_{ki} } \right)} = \sum\limits_{k = 1}^n {\left( {a_{ik} } \right)\left( {a_{ik} } \right)} = \sum\limits_{k = 1}^n {\left( {a_{ik} } \right)^2 }$ .
But that can only happen if and only if $\left( {\forall i,k} \right)\left[ {a_{ik} = 0} \right]$ .