Solve this

Given:

$A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]$

$A(\operatorname{adj} A)=\left[\begin{array}{ll}k & 0 \\ 0 & k\end{array}\right]$

Now,

$A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]$

$\Rightarrow|A|=\left|\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right|$

$\Rightarrow|A|=\cos ^{2} x+\sin ^{2} x$

$\Rightarrow|A|=1$

As we know,

$A(\operatorname{adj} A)=|A| I$

$\Rightarrow\left[\begin{array}{ll}k & 0 \\ 0 & k\end{array}\right]=|A| I$

$\Rightarrow k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=|A| I$

$\Rightarrow k I=|A| I$

$\Rightarrow k=|A|$

$\Rightarrow k=1 \quad(\because|A|=1)$

Hence, $k=1$