Let S be the set containing all 3 × 3 matrices with entries from {-1, 0, 1}. The total number of matrices A $\in$ S such that the sum of all the diagonal elements of A^T A is 6 is __________

$$\begin{gathered} \mathrm{Tr}\left({\mathrm{AA}}^{\mathrm{T}}\right)=6 \\ {\mathrm{AA}}^{\mathrm{T}}=\left[\begin{array}{l}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{d}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{g}\\ \mathrm{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{e}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{h}\\ \mathrm{c}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{f}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{i}\end{array}\right]\left[\begin{array}{l}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{c}\\ \mathrm{d}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{e}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{f}\\ \mathrm{g}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{h}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{i}\end{array}\right] \end{gathered}$$

Now given ${\mathrm{a}}^2+{\mathrm{d}}^2+{\mathrm{g}}^2+{\mathrm{b}}^2+{\mathrm{e}}^2+{\mathrm{h}}^2+{\mathrm{c}}^2+{\mathrm{f}}^2+{\mathrm{i}}^2=6$

three elements must be equal to zero and other 6 elements are 1,-1

$\begin{array}{l}{=}^9{\mathrm{C}}_3\times 2^6\\ =5376\end{array}$