Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
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 nonzero real numbers $a_1, a_2, a_3, \ldots, a_9$ then $k = (a+b)!$ where $\gcd(a,b) = 1$ equals
Step-by-Step Solution
Key Concept: A third-order determinant has 9 positions that can be filled with 9 distinct nonzero real numbers in 9! ways. The problem requires expressing 9! as (a+b)! where gcd(a,b)=1, leading to a+b=9 as the unique solution.
The number of third-order determinants equals the number of arrangements of nine different numbers in places, which is $9!$. The constraint is $(a+b)! = 9!$, so $a+b = 9$.
Correct Answer: 9