$$\begin{gathered} \left( \begin{array}{cc}2 & 3\\ 5 & 7\end{array} \right) \left( \begin{array}{cc} 1& - 3\\ - 2& 4\end{array} \right) = \left( \begin{array}{cc}- 4& 6\\ - 9& \mathrm{x}\end{array} \right) \\ \Rightarrow \left( \begin{array}{cc}- 4& 6\\ - 9& 13\end{array} \right) = \left( \begin{array}{cc}- 4& 6\\ - 9& \mathrm{x}\end{array} \right) \end{gathered}$$

By equality of matrices,

x = 13.

According to the given operation

$5 * 7 = \mathrm{L}.\mathrm{C}.\mathrm{M}. ( 5, 7 )= 35$

$$\begin{gathered} \mathrm{Principal} \mathrm{value} \mathrm{of} {\cos }^{-1} \left( \frac{1}{2} \right) = \frac{\mathrm{\pi }}{3} \\ \mathrm{Principal} \mathrm{value} \mathrm{of} {\sin }^{-1} \left( \frac{- 1}{2} \right) = \frac{- \mathrm{\pi }}{6} \\ \mathrm{Hence} \mathrm{principal} \mathrm{value} \mathrm{of} {\cos }^{-1} \left( \frac{1}{2} \right) - 2 {\sin }^{-1} \left( \frac{- 1}{2} \right) \\ = \frac{\mathrm{\pi }}{3} - 2 \left(\frac{- \mathrm{\pi }}{6} \right) \\ = \frac{2\mathrm{\pi }}{3}. \end{gathered}$$

$$\begin{gathered} \mathrm{coc} \mathrm{\theta } \left[ \begin{array}{cc}\cos \mathrm{\theta } & \sin \mathrm{\theta }\\ - \sin \mathrm{\theta } & \cos \mathrm{\theta }\end{array} \right] + \sin \mathrm{\theta } \left[ \begin{array}{cc}\sin \mathrm{\theta } & - \cos \mathrm{\theta } \\ \cos \mathrm{\theta } & \sin \mathrm{\theta }\end{array}\right] \\ = \left[ \begin{array}{cc} {\cos }^2 \mathrm{\theta } & \cos \mathrm{\theta } \sin \mathrm{\theta }\\ - \sin \mathrm{\theta } \cos \mathrm{\theta } & {\cos }^2 \mathrm{\theta }\end{array} \right] + \left[ \begin{array}{cc}{\sin }^2 \mathrm{\theta } & - \sin \mathrm{\theta } \cos \mathrm{\theta } \\ \sin \mathrm{\theta } \cos \mathrm{\theta } & {\sin }^2 \mathrm{\theta }\end{array}\right] \\ = \left[ \begin{array}{cc} {\cos }^2 \mathrm{\theta } + {\sin }^2 \mathrm{\theta } & \cos \mathrm{\theta } \sin \mathrm{\theta } - \sin \mathrm{\theta } \cos \mathrm{\theta }\\ - \sin \mathrm{\theta } \cos \mathrm{\theta } + \cos \mathrm{\theta } \sin \mathrm{\theta } & {\cos }^2 \mathrm{\theta } + {\sin }^2 \mathrm{\theta } \end{array} \right] \\ = \left[ \begin{array}{cc}1 & 0\\ 0 & 1\end{array} \right] ..........\left[ \because {\cos }^2 \mathrm{\theta } + {\sin }^2 \mathrm{\theta } = 1 \right] \end{gathered}$$

$\mathrm{Minor} \mathrm{of} \mathrm{the} \mathrm{element} {\mathrm{a}}_{32} = \left| \begin{array}{cc}5& 8\\ 2& 1\end{array} \right|$

                                  = 5 - 16

                                  = - 11