Question

The determinant $ \begin{vmatrix} \cos (\alpha +\beta ) & -\sin (\alpha +\beta ) & \cos 2\beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \\ \end{vmatrix} $ is independent of

• A. $ \alpha $
• B. $ \beta $
• C. $ \alpha $ and $ \beta $
• D. Neither $ \alpha $ nor $ \beta $

Answer 1

Answer: (A)

We have the determinant
$\begin{vmatrix} \cos (\alpha +\beta ) & -\sin (\alpha +\beta ) & \cos 2\beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \\ \end{vmatrix} $
$ {{R}_{1}}\to {{R}_{1}}+(sin\beta ){{R}_{2}}+(cos\beta ){{R}_{3}}, $
$\begin{vmatrix} 0 & 0 & 1+\cos 2\beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \\ \end{vmatrix} $
$ =(1+\cos 2\beta )\,({{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha ) $
$ =1+\cos 2\beta $
which is independent of $ \alpha $ .