If \(A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & -\cos \theta\end{array}\right]\), then \(A^{-1}=\)

We have \(A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & -\cos \theta\end{array}\right] \Rightarrow \operatorname{adj} A=\left[\begin{array}{cc}-\cos \theta & \sin \theta \\ \sin \theta & \cos \theta\end{array}\right]\)
\(|A|=\left|\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & -\cos \theta\end{array}\right|=-\cos ^{2} \theta-\sin ^{2} \theta=-\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=-1\)
\(\therefore A^{-1}=\frac{1}{(-1)}\left[\begin{array}{cc}-\cos \theta & \sin \theta \\ \sin \theta & \cos \theta\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & -\cos \theta\end{array}\right]\)