Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
If $S_r = a^r + b^r + c^r$ the value of $\begin{vmatrix} S_0 & S_1 & S_2 \\ S_1 & S_2 & S_3 \\ S_2 & S_3 & S_4 \end{vmatrix}$ is equal to $(\alpha - \beta)^2(\beta - \gamma)^{2k-2}(\gamma - a)^{k^2-2}$. Then $k$ is
Step-by-Step Solution
Key Concept: Recognizing the Vandermonde determinant form immediately yields the factorized answer without full expansion.
The matrix has entries $S_i = \sum_{j=0}^{i} α^j β^j γ^j$. The determinant is expressed using Vandermonde structure: $\begin{vmatrix} 1 & 1 & 1 \\ α & β & γ \\ α^2 & β^2 & γ^2 \end{vmatrix}^2 = (α-β)^2(β-γ)^2(γ-α)^2$, which represents the product of all pairwise differences squared.
Correct Answer: 2