Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade None
Question:
Let $\{\Delta_1, \Delta_2, \Delta_3, \ldots \Delta_k\}$ be the set of third order determinants that can be made with the distinct non-zero real numbers $a_1, a_2, \ldots a_9$. Then:
$k = 9!$
$\sum_{i=1}^{k} \Delta_i = 0$
At least one $\Delta_i = 0$
None of these
Step-by-Step Solution
Key Concept: Interchange of rows or columns changes the sign of a determinant; pairing such determinants causes their sum to vanish.
The number of third-order determinants using nine different numbers in nine positions equals the number of permutations of nine distinct objects, which is $9!$. For each determinant formed, interchanging two consecutive rows or columns produces a related determinant with opposite sign. The sum of pairs of determinants obtained by such interchanges equals zero, as each determinant and its negation cancel.
Correct Answer: 1,2