Question

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

Answer 1

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

On expanding along first row, we get

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

$\Rightarrow \Delta =-sin\alpha (0-sin\beta cos\alpha )-cos\alpha \left(sin\alpha sin\beta \right)$

$\Rightarrow \Delta =sin\alpha cos\alpha sin\beta -sin\alpha cos\alpha sin\beta$

$\Rightarrow \Delta =0$