Matrices & Determinants
Properties of Determinants
Grade 12
Question:
<p>If \[f(x) = \begin{vmatrix} \cos(x+\alpha) & \cos(x+\beta) & \cos(x+\gamma) \\ \sin(x+\alpha) & \sin(x+\beta) & \sin(x+\gamma) \\ \sin(\beta-\gamma) & \sin(\gamma-\alpha) & \sin(\alpha-\beta) \end{vmatrix}\] and \(f(0) = -2\), then find the value of \(\displaystyle\sum_{r=1}^{30} |f(r)|\).</p>
Step-by-Step Solution
Key Concept: The determinant f(x) is independent of x (constant for all x) because the third row contains fixed angle differences. Once you find f(0), you can determine the constant value, then compute the sum directly.
<p><strong>Step 1: Prove f(x) is constant</strong></p><p>Expand the determinant along the third row. Each cofactor involves a 2×2 determinant of the form:</p><p>∣cos(x+a) cos(x+b)∣ = cos(x+a)sin(x+b) - cos(x+b)sin(x+a) = sin(b-a)</p><p>∣sin(x+a) sin(x+b)∣</p><p>This is independent of x. Therefore, f(x) = constant for all x.</p><p><strong>Step 2: Find the constant using f(0)</strong></p><p>Given: f(0) = -2</p><p>Since f(x) is constant, f(r) = f(0) = -2 for all r ∈ ℝ</p><p><strong>Step 3: Calculate the sum</strong></p><p>∑(r=1 to 30) |f(r)| = ∑(r=1 to 30) |−2| = ∑(r=1 to 30) 2 = 2 × 30 = 60</p><p><strong>∴ Answer: 60</strong></p>
Correct Answer: 60