Evaluate $\Delta=\left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$ Solution Expanding along $\mathrm{R}_{1}$, we get $$\begin{aligned} \Delta & =0\left|\begin{array}{cc} 0 & \sin \beta \\ -\sin \beta & 0 \end{array}\right|-\sin \alpha\left|\begin{array}{cc} -\sin \alpha & \sin \beta \\ \cos \alpha & 0 \end{array}\right|-\cos \alpha\left|\begin{array}{cc} -\sin \alpha & 0 \\ \cos \alpha & -\sin \beta \end{array}\right| \\ & =0-\sin \alpha(0-\sin \beta \cos \alpha)-\cos \alpha(\sin \alpha \sin \beta-0) \\ & =\sin \alpha \sin \beta \cos \alpha-\cos \alpha \sin \alpha \sin \beta=0 \end{aligned}$$

Step 1: Expand along the first row

To evaluate the determinant of a $$3 \times 3$$ matrix, we can expand along any row or column. Here, we choose to expand along the first row ( $$R_1$$ ). The formula for expansion along $$R_1$$ is $$a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$ , where $$a_{ij}$$ are the elements and $$C_{ij}$$ are their cofactors. The cofactors include a sign factor $$(-1)^{i+j}$$ .

Step 2: Calculate the 2x2 determinants

Now, we evaluate each of the $$2 \times 2$$ determinants. The determinant of a $$2 \times 2$$ matrix $$\left|\begin{array}{cc}a & b \\ c & d\end{array}\right|$$ is given by ad - bc. We apply this formula to each of the three $$2 \times 2$$ sub-determinants.

Step 3: Simplify the expression

We simplify the terms inside the parentheses. The first term becomes $$0(0 + \sin^2 \beta) = 0$$ . The second term simplifies to $$-\sin \alpha(-\sin \beta \cos \alpha)$$ . The third term simplifies to $$-\cos \alpha(\sin \alpha \sin \beta)$$ .

Step 4: Combine and find the final value

After simplifying, we get two terms: $$\sin \alpha \sin \beta \cos \alpha$$ and $$-\cos \alpha \sin \alpha \sin \beta$$ . These two terms are identical but with opposite signs.

Step 5: Final Result

Since the two terms are identical with opposite signs, they cancel each other out, resulting in a final determinant value of $$0$$ .

More from this chapter

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