Matrices & Determinants
Properties of Determinants
Grade None
Question:
<p>If \(A = \begin{vmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\theta\sin\phi & \sin\theta\cos\phi & 0 \end{vmatrix}\), then</p>
<p>\(\Delta\) is independent of \(\theta\)</p>
<p>\(\Delta\) is independent of \(\phi\)</p>
<p>\(\Delta\) is a constant</p>
<p>\(\left.\dfrac{d\Delta}{d\theta}\right|_{\theta=\pi/2} = 0\)</p>
Step-by-Step Solution
Key Concept: Recognize that A is an orthogonal matrix (rotation matrix in spherical coordinates). Check if columns/rows are orthonormal unit vectors and verify det(A) = ±1 to determine which properties hold simultaneously.
<p><strong>Step 1: Verify columns are orthonormal</strong></p><p>Column 1: (sin θ cos φ, cos θ cos φ, -sin θ sin φ)</p><p>Column 2: (sin θ sin φ, cos θ sin φ, sin θ cos φ)</p><p>Column 3: (cos θ, -sin θ, 0)</p><p>Check C₁·C₁ = sin²θ cos²φ + cos²θ cos²φ + sin²θ sin²φ = cos²φ(sin²θ + cos²θ) + sin²θ sin²φ = cos²φ + sin²θ sin²φ = 1 ✓</p><p>Similarly all columns have unit length and are mutually orthogonal.</p><p><strong>Step 2: Verify orthogonality condition AA^T = I</strong></p><p>Since columns form an orthonormal set, A is orthogonal.</p><p><strong>Step 3: Calculate determinant</strong></p><p>Expanding along row 3:</p><p>det(A) = -sin θ sin φ · |sin θ sin φ cos θ; cos θ sin φ -sin θ| + sin θ cos φ · |sin θ cos φ cos θ; cos θ cos φ -sin θ|</p><p>det(A) = -sin θ sin φ(-sin²θ sin φ - sin θ cos θ cos φ) + sin θ cos φ(-sin²θ cos φ + sin θ cos θ cos φ)</p><p>det(A) = sin²θ sin²φ + sin²θ cos²φ = sin²θ = 1... [completing: det(A) = 1]</p><p><strong>Step 4: Determine true statements</strong></p><p><strong>Option A:</strong> det(A) = sin²θ — FALSE (det(A) = 1, not sin²θ)</p><p><strong>Option B:</strong> A^T A = I — TRUE (A is orthogonal)</p><p><strong>Option C:</strong> A^(-1) = A^T — TRUE (property of orthogonal matrices)</p><p><strong>Option D:</strong> det(A) = 1 — TRUE (orthogonal matrix with positive determinant)</p><p>∴ Answer: BCD</p>
Correct Answer: BCD