The reaction is as follows:
\[ K_2\text{Cr}_2\text{O}_7 + 4\text{KCl} + 6\text{H}_2\text{SO}_4 \rightarrow 2\text{CrO}_2\text{Cl}_2 + 6\text{KHSO}_4 + 3\text{H}_2\text{O} \]This reaction is known as the chromyl chloride test. In this reaction, the oxidation state of chromium in \(\text{CrO}_2\text{Cl}_2\) is \(+6\).
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.
Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to: