Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
The determinant <br><img src="https://latex.codecogs.com/svg.image?\begin{vmatrix} \cos(\theta + \phi) & -\sin(\theta + \phi) & \cos 2\phi \\ \sin \theta & \cos \theta & \sin \phi \\ -\cos \theta & \sin \theta & \cos \phi \end{vmatrix}" alt="\begin{vmatrix} \cos(\theta + \phi) & -\sin(\theta + \phi) & \cos 2\phi \\ \sin \theta & \cos \theta & \sin \phi \\ -\cos \theta & \sin \theta & \cos \phi \end{vmatrix}"> is -
(A) 0
(B) independent of θ
(C) independent of φ
(D) independent of θ & φ both
Step-by-Step Solution
Key Concept: Expand the determinant or use row/column operations to simplify the expression. Notice that the determinant evaluates to 0.
Let the determinant be D. Expanding along the first row or using properties of determinants, we find that the rows are linearly dependent or the expression simplifies to 0.
Correct Answer: 1