Sum of all diagonal elements is equal to sum of square of each element of the matrix.

i.e., $$A = \left[ {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{b_1}} & {{b_2}} & {{b_3}} \cr {{c_1}} & {{c_2}} & {{c_3}} \cr } } \right]$$

then $${t_r}\,(A\,.\,{A^T})$$

$$ = a_1^2 + a_2^2 + a_3^2 + b_1^2 + b_2^2 + b_3^2 + c_1^2 + c_2^2 + c_3^2$$

$$\because$$ $${a_i},{b_i},{c_i} \in \{ - 1,0,1\} $$ for $$i = 1,2,3$$

$$\therefore$$ Exactly three of them are zero and rest are 1 or $$-$$1.

Total number of possible matrices $${}^9{C_3} \times {2^6}$$

$$ = {{9 \times 8 \times 7} \over 6} \times 64$$

$$ = 5376$$