Matrices & Determinants
Sum of Determinants — Finding n
nta_pyq_2023_apr
Grade 12
Question:
Let $D_k=\begin{vmatrix}1&2k&2k-1\\n&n^2+n+2&n^2\\n&n^2+n&n^2+n+2\end{vmatrix}$. If $\displaystyle\sum_{k=1}^n D_k=96$, then $n$ is equal to _________.
Step-by-Step Solution
Key Concept: Compute $\sum D_k$ by summing column-1 across $k=1$ to $n$, noting $\sum 1=n$, $\sum 2k=n(n+1)$, $\sum(2k-1)=n^2$. Then expand the resulting determinant.
$2n(n+2)=96\Rightarrow n^2+2n=48\Rightarrow n=6$.
Correct Answer: 6