Let S be the set containing all 3 × 3 matrices with entries from - 1 , 0 , 1 . The total number of…

Given that the sum of all the diagonal elements of $A^{T}A$ is $6$, means sum of all diagonal elements is equal to sum of square each element of the matrix.

i.e., $A=\left[\begin{array}{lll}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]$ and $A^T=\left[\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$

Then $t_r\left(A·A^T\right)$$=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$

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

Now sum to be $6$ exactly three of them should be zero and rest should be $1$ or $-1$.

So, total number of possible matrices ${}^{9}C_3\times 2^6$

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

$=5376$