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Let $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & -\cos \theta\end{array}\right]$, then the inverse of $A$
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Verified Answer
The correct answer is:
$\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & -\cos \theta\end{array}\right]$
$$
\begin{array}{l}
\quad|A|=\left|\begin{array}{cc}
\cos \theta & -\sin \theta \\
-\sin \theta & -\cos \theta
\end{array}\right|=-1 \\
\operatorname{adj}(A)=\left[\begin{array}{cc}
-\cos \theta & \sin \theta \\
\sin \theta & \cos \theta
\end{array}\right] \\
\therefore \quad A^{-1}=\left|\begin{array}{cc}
\cos \theta & -\sin \theta \\
-\sin \theta & -\cos \theta
\end{array}\right|=A
\end{array}
$$
\begin{array}{l}
\quad|A|=\left|\begin{array}{cc}
\cos \theta & -\sin \theta \\
-\sin \theta & -\cos \theta
\end{array}\right|=-1 \\
\operatorname{adj}(A)=\left[\begin{array}{cc}
-\cos \theta & \sin \theta \\
\sin \theta & \cos \theta
\end{array}\right] \\
\therefore \quad A^{-1}=\left|\begin{array}{cc}
\cos \theta & -\sin \theta \\
-\sin \theta & -\cos \theta
\end{array}\right|=A
\end{array}
$$
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