If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$ and $A(\operatorname{adj} A=)\left[\begin{array}{cc}k & 0 \\ 0 & k\end{array}\right]$, then find the value of $k$.
$A=\left[\begin{array}{ll}\cos \theta & \sin \theta\end{array}\right.$
$-\sin \theta \quad \cos \theta]$
$\therefore|A|=\mid \cos \theta \quad \sin \theta$
$-\sin \theta \quad \cos \theta \mid=\cos ^{2} \theta+\sin ^{2} \theta=1 \neq 0$
Thus, $A^{-1}$ exists.
Now,
$A^{-1}=\frac{\operatorname{adj} A}{|A|}=\left[\begin{array}{ll}\cos \theta & -\sin \theta\end{array}\right.$
$\sin \theta \quad \cos \theta]$
$\Rightarrow A^{-1}=\operatorname{adj} A$
$\Rightarrow A A^{-1}=A \operatorname{adj} A$
$\Rightarrow A A^{-1}=\left[\begin{array}{ll}k & 0\end{array}\right.$
$0 \mathrm{~A}]=\left[\begin{array}{ll}k & 0\end{array}\right.$
$\left.\begin{array}{ll}0 & k\end{array}\right] \quad\left[\because A A^{-1}=I\right]$
$\Rightarrow k=1$
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