Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
Given, $a^n b^{2n-2}$ where $\begin{vmatrix} 1 & a^2 & a^4 \\ 1 & b^2 & b^4 \\ 1 & c^2 & c^4 \end{vmatrix}$ and on comparison, we get $n = -2$
Step-by-Step Solution
Key Concept: Vandermonde determinant properties allow factorization of determinants with powers of variables
The determinant $\begin{vmatrix} 1 & a^2 & a^4 \\ 1 & b^2 & b^4 \\ 1 & c^2 & c^4 \end{vmatrix}$ can be factored using the Vandermonde determinant pattern. Expanding and simplifying the given expression $a^n b^{2n-2}$ by comparing coefficients with the determinant expansion gives $n = -2$.
Correct Answer: -2