Let $\{D_1,D_2,D_3,\mathrm{.........}{D}_n\}$ be the set of third order determinant that can be made with the distinct non-zero real numbers $a_1,a_2\mathrm{......}{a}_9$ (using all), then

Total number of third order determinants is 9! Since number of determinants are even and in which there are $\frac{9!}{2}$ pairs of determinants which are obtained by changing two consecutive rows. So, ${\displaystyle \sum D_i=0.}$