Question

The value of the determinant $\left|\begin{array}{ccc}cos\alpha & -sin\alpha & 1\\ sin\alpha & cos\alpha & 1\\ cos(\alpha +\beta )& -sin(\alpha +\beta )& 1\end{array}\right|$ is

• A. Independent of $\alpha$
• B. Independent of $\beta$
• C. Independent of $\alpha$ and $\beta$
• D. None of the above

Answer 1

Answer: (A)

Independent of $\alpha$

Given, $\left|\begin{array}{ccc}cos\alpha & sin\alpha & 1\\ sin\alpha & cos\alpha & 1\\ cos(\alpha +\beta )& -sin(\alpha +\beta )& 1\end{array}\right|$

[Applying $R_3\to R_3-R_1\left(cos\beta \right)+R_2\left(sin\beta \right)$]

$\left|\begin{array}{ccc}cos\alpha & sin\alpha & 1\\ sin\alpha & cos\alpha & 1\\ 0& 0& 1+sin\beta -cos\beta \end{array}\right|$

$=(1+sin\beta -cos\beta )(co{s}^2\alpha +si{n}^2\alpha )$

$=1+sin\beta -cos\beta$, which is independent of $\alpha$.

Hence, option A is correct.