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If \(\Delta_r\) denotes the determinant \[ \Delta_r=\begin{vmatrix} \frac{r}{n} & \frac{2r-1}{n-1} & \frac{3r-2}{a} \\ \frac12 n(n-1) & \frac12 (n-1)^2 & \frac12 (n-1)(3n+4) \end{vmatrix}, \] then the sum \(\sum_{r=1}^{n-1} \Delta_r\) is:

1. independent of both \(a\) and \(n\)
2. dependent only on \(a\)
3. dependent only on \(n\)
4. dependent on both \(a\) and \(n\)

Correct answer: independent of both \(a\) and \(n\)

Solution

The determinant \\( ext{det}( ext{matrix})\\ ext{ is computed in such a way that the variables } a ext{ and } n ext{ do not affect the overall sum } \\ ext{as the contributions from each } r ext{ cancel out, leading to a result that remains constant regardless of these parameters.}

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