Evaluate = | ccc 0 & & - - & 0 & & - & 0 | - Vidyadip

Question

Evaluate $\Delta=\left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$

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Answer

$\Delta =0 \left|{\begin{array}{*{20}{c}} 0&{\sin \beta } \\ { - \sin \beta }&0 \end{array}} \right| - \sin \alpha \left|{\begin{array}{*{20}{c}} { - \sin \alpha }&{\sin \beta } \\ {\cos \alpha }&0 \end{array}} \right|$$ - \cos \alpha \left| {\begin{array}{*{20}{c}} { - \sin \alpha }&0 \\ {\cos \alpha }&{ - \sin \beta } \end{array}} \right|$
$=0-sin\alpha(0-\cos\alpha\sin\beta)-\cos\alpha(\sin\alpha\sin\beta)$
$=\sin\alpha\cos\alpha\sin\beta-\cos\alpha\sin\alpha\sin\beta =0 $

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