Matrices & Determinants
Matrices & Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
$A = [a_{ij}]_{n \times n}$ be a square matrix, $n$ is odd such that $a_{ij} = (-1)^j C_j^n C_i^n$, then trace $(A) = $ (Trace $(A)$ denotes sum of diagonal elements of $A$)
Step-by-Step Solution
Key Concept: The trace is invariant under similarity transformations and can be related to combinatorial identities involving binomial coefficients.
The trace of $A$ equals $\sum_{i=1}^n a_{ii}$ and can also be written as $\sum_{i=0}^n (-1)^i (^nC_i)^2$ using binomial coefficients. For an $n \times n$ matrix, this alternating sum of squared binomial coefficients equals $(-1)^{(n-1)/2}(^nC_{(n-1)/2})^2 = -1$ when $n$ is odd, as shown by the identity for binomial sums.
Correct Answer: 3