Question

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

• A. $4\cos \alpha \cos \beta \cos \gamma$
• B. $2\cos \alpha \cos \beta \cos \gamma$
• C. $4\sin \alpha \sin \beta \sin \gamma$
• D. none of these

Answer 1

The correct option is D none of these

$\left|\begin{array}{ccc}1& \cos (\beta -\alpha )& \cos (\gamma -\alpha )\\ \cos (\alpha -\beta )& 1& (\cos \gamma -\beta )\\ \cos (\alpha -\gamma )& \cos (\beta -\gamma )& 1\end{array}\right|$
$=\left|\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ \cos \beta & \sin \beta & 0\\ \cos \gamma & \sin \gamma & 0\end{array}\right|\left|\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ \cos \beta & \sin \beta & 0\\ \cos \gamma & \sin \gamma & 0\end{array}\right|$ (row by row multiplication)
$=0$
Hence, option D.