\[
D_{r_{1}, r_{2}}=\left[\begin{array}{c}
\tan ^{-1} \frac{\eta}{r_{2}}-4 \\
\cot ^{-1} \frac{\eta}{r_{2}}
\end{array}\right] \text { if } \Delta_{n}=\sum_{\eta=1}^{n} \sum_{z^{=1}}^{n} D_{n}
\]
Consider a square matrix of order 2 given by
then find the determinant value of the matrix \( A \) defined by \( A=\operatorname{Lim}_{n \rightarrow \infty} \frac{\Delta_{n}}{n^{3} \sin \left(\frac{\sqrt{\pi}}{n}\right.} \).
