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Determinants

Example 4

Evaluate \(\Delta=|\begin{smallmatrix}0&sin~\alpha&-cos~\alpha\\ -sin~\alpha&0&sin~\beta\\ cos~\alpha&-sin~\beta&0\end{smallmatrix}|.\)

Solution Expanding along \(R_{1},\) we get DETERMINANTS 81 \(\Delta=0|\begin{smallmatrix}0&sin~\beta\\ -sin~\beta&0\end{smallmatrix}|-sin~\alpha|\begin{smallmatrix}-sin~\alpha&sin~\beta\\ cos~\alpha&0\end{smallmatrix}|-cos~\alpha|\begin{smallmatrix}-sin~\alpha&0\\ cos~\alpha&-sin~\beta\end{smallmatrix}|\) \(=0-sin~\alpha(0-sin~\beta~cos~\alpha)-cos~\alpha(sin~\alpha~sin~\beta-0)\) \(=sin~\alpha~sin~\beta~cos~\alpha-cos~\alpha~sin~\alpha~sin~\beta=0\)