Let $\mathrm{S}=\left\{\left[\begin{array}{c}\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\end{array}\right]: a_{ij}\in \left\{-1,0,1\right\}\right\}$, then the number of symmetric matrices with trace equals zero, is

In a third order matrix $A, a_{ij}$ denotes the element in the $i^{\mathrm{th}}$ row and $j^{\mathrm{th}}$ column.

If $a_{ij}=0$ for $i=j$
        $=1$ for $i>j$
        $=-1$ for $i<j$

Then the matrix is

Let $A=\left[a_{ij}\right]$ is a square matrix of order $2$ where $a_{ij}=i^2-j^2$. Then $A$ is