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Let $\{D_1, D_2, D_3,...., D_n\}$ be the set of all third order determinants that can be made with the distinct non-zero real numbers $a_1, a_2,...., a_n$, then |
$\sum\limits_{i=1}^{n}D_i=1$ $\sum\limits_{i=1}^{n}D_i=0$ $D_i =D_j$ for all i, j none of these |
$\sum\limits_{i=1}^{n}D_i=0$ |
Total number of third order determinants with distinct non-zero real numbers $a_1, a_2,...., a_n$ as elements is 9!. These determinants can be grouped into two groups each containing $\frac{9!}{2}$ 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, $\sum\limits_{i=1}^{n}D_i=0$ |
