Question

The determinant
$\left|\begin{array}{ccc}cos(\alpha +\beta )& -sin(\alpha +\beta )& cos2\beta \\ sin\alpha & cos\alpha & sin\beta \\ -cos\alpha & sin\alpha & cos\beta \end{array}\right|$ is independent of

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

Answer 1

Answer: (A)

$\alpha$

Upon solving the matrix ,

$\left|\begin{array}{ccc}cos(\alpha +\beta )& -sin(\alpha +\beta )& cos2\beta \\ sin\alpha & cos\alpha & sin\beta \\ -cos\alpha & sin\alpha & cos\beta \end{array}\right|$

$=$ $cos(\alpha +\beta )\times (cos\alpha \times cos\beta -sin\alpha \times sin\beta )$ + $sin(\alpha +\beta )$ $\times (sin\alpha \times cos\beta +sin\beta \times cos\alpha )$ $+cos2\beta \times (si{n}^2\alpha +co{s}^2\alpha ))$

$=co{s}^2(\alpha +\beta )+si{n}^2(\alpha +\beta )+cos2\beta$

$=cos2\beta$

Thus the final result is independent of $\alpha$

Thus the correct answer is A.