Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
If $A = \begin{vmatrix} \sin\theta\cos\theta & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\phi\sin\phi & \sin\theta\sin\phi & 0 \end{vmatrix}$ then
$\Delta$ is independent of $\theta$
$\Delta$ is independent of $\phi$
$\Delta$ is a constant
$\frac{d\Delta}{d\theta}\bigg|_{\theta=\frac{\pi}{2}} = 0$
Step-by-Step Solution
Key Concept: The determinant $\Delta$ can be simplified using column/row operations to reveal its dependence on $\theta$ and $\phi$. Computing $\frac{d\Delta}{d\theta}$ and evaluating at specific angles tests whether the determinant reduces to a constant or angle-dependent expression.
Applying column operation $C_1 \to C_1 - (\cot\phi)C_2$ to the determinant $\Delta$, the determinant becomes $\frac{\sin 0}{\sin\phi}[-\sin\phi\sin^2\theta - \cos^2\theta\sin\phi]$. Expanding along $C_1 = \sin 0$ yields a result independent of $\phi$. Computing $\frac{d\Delta}{d\theta} = \cos 0$ gives $\left[\frac{d\Delta}{d\theta}\right]_{\theta=\pi/2} = \cos\frac{N}{2} = 0$.
Correct Answer: 2,4