Matrices & Determinants
Sum of Determinants
nta_pyq_2024_apr
Grade 12
Question:
For $\alpha,\beta\in\mathbb{R}$ and a natural number $n$, let $A_r=\begin{vmatrix}r & 1 & \frac{n^2}{2}+\alpha\\ 2r & 2 & n^2-\beta\\ 3r-2 & 3 & \frac{n(3n-1)}{2}\end{vmatrix}$. Then $\displaystyle\sum_r A_r$ is:
0
$4\alpha+2\beta$
$2\alpha+4\beta$
$2n$
Step-by-Step Solution
Key Concept: Compute $2A_{10}-A_8$ to extract a determinant independent of $r$, then simplify.
$\sum A_r=4\alpha+2\beta$.
Correct Answer: 2