Let `{D_1, D_2, D_3 ,D_n}` be the set of third order determinant that can be made with the distinct non-zero real numbers `a_1, a_2, a_qdot` Then `sum_(i=1)^n D_i=1` b. `sum_(i=1)^n D_i=0` c. `D_i-D_j ,AAi ,j` d. none of these

Options

A. `underset(i=1)overset(n)sum D_(i) =1`
B. `underset(i=1)overset(n) D_(1) =0`
C. `D_(i) = D_(j)` for all i, j
D. none of these

Answer

B

Explanation

Total number of third order determinants with distinct non-zero real numbers `a_(1), a_(2),..,a_(9)` as elements is 9!. These determinants can be grouped into two groups each containing `(9!)/(2)` determinants such that corresponding to each determinant in a group there is another determinant in the other group which is obtained by interchanging two consecutive rows of the determinant in first group.
Hence, `underset(i=1)overset(n)sum D_(i) =0`