$\text{ The number of all }3\times 3\text{ matrices }\mathrm{A}\text{ , with entries from the set }\{-1,0,1\}\text{ such that the }$$\text{ sum of the diagonal elements of }{\mathrm{AA}}^{\mathrm{T}}\text{ is }3\text{ . is }$

$\begin{array}{l}tr\left({\mathrm{AA}}^{\mathrm{T}}\right)=3\\ a_{11}^2+a_{12}^2+a_{13}^2+a_{21}^2+\dots \dots \dots +a_{33}^2=3\mid \\ \text{ Possible cases }\\ \left.\begin{array}{cl}0,0,0,0,0,0,1,1,1& \to 1\\ 0,0,0,0,0,0,-1,-1,-1& \to 1\\ 0,0,0,0,0,01,1,-1 & \to 3\\ 0,0,0,0,0,0,-1,1,-1& \to 3\end{array}\right\}{}^{9}C_6\times 8=84\times 8=672\end{array}$