Let {D1,D2,D3,.........Dn} be the set of third order determinant that can be made with the distinct non-zero real numbers a1,a2......  a9 (using all), then

Total number of third order determinants is 9! Since number of determinants are even and in which there are 9!2 pairs of determinants which are obtained by changing two consecutive rows. So, ∑Di=0.