Matrices & Determinants
Determinants
Grade 12
Question:
<p>If <math>a, b, \gamma \in \mathbb{R}</math>, then the determinant <math>D = \begin{vmatrix} (e^{i\alpha} + e^{-i\alpha})^2 & (e^{i\alpha} - e^{-i\alpha})^2 & 4 \\ (e^{i\beta} + e^{-i\beta})^2 & (e^{i\beta} - e^{-i\beta})^2 & 4 \\ (e^{i\gamma} + e^{-i\gamma})^2 & (e^{i\gamma} - e^{-i\gamma})^2 & 4 \end{vmatrix}</math> is</p>
<p>(a) independent of <math>\alpha, \beta</math> and <math>\gamma</math></p>
<p>(b) dependent on <math>\alpha, \beta</math> and <math>\gamma</math></p>
<p>(c) independent of <math>\alpha, \beta</math></p>
<p>(d) independent of <math>\alpha, \gamma</math></p>
Step-by-Step Solution
Key Concept: Simplify each entry using exponential identities: (e^{iθ} + e^{-iθ})² = 4cos²θ and (e^{iθ} - e^{-iθ})² = -4sin²θ, then observe the resulting determinant has two identical columns.
<p><strong>Step 1: Simplify the first column entries</strong></p><p>For any angle θ, using Euler's formula e^{iθ} = cosθ + isinθ:</p><p>(e^{iθ} + e^{-iθ})² = (2cosθ)² = 4cos²θ</p><p>So column 1 entries: 4cos²α, 4cos²β, 4cos²γ</p><p><strong>Step 2: Simplify the second column entries</strong></p><p>(e^{iθ} - e^{-iθ})² = (2i·sinθ)² = 4i²sin²θ = -4sin²θ</p><p>So column 2 entries: -4sin²α, -4sin²β, -4sin²γ</p><p><strong>Step 3: Rewrite the determinant with simplified entries</strong></p><p>D = $\begin{vmatrix} 4cos²α & -4sin²α & 4 \\ 4cos²β & -4sin²β & 4 \\ 4cos²γ & -4sin²γ & 4 \end{vmatrix}$</p><p><strong>Step 4: Use the trigonometric identity cos²θ + sin²θ = 1</strong></p><p>Note that cos²θ - sin²θ = cos(2θ), so: 4cos²θ - 4sin²θ = 4(cos²θ + sin²θ) - 8sin²θ = 4 - 8sin²θ</p><p>But more directly, observe: 4cos²θ + 4sin²θ = 4(cos²θ + sin²θ) = 4</p><p><strong>Step 5: Add Column 1 and Column 2</strong></p><p>Column 1 + Column 2 gives: [4cos²α - 4sin²α, 4cos²β - 4sin²β, 4cos²γ - 4sin²γ]ᵀ = [4(cos²θ - sin²θ)]ᵀ</p><p>However, notice: 4cos²θ + (-4sin²θ) = 4cos²θ - 4sin²θ</p><p>By the identity cos²θ - sin²θ = (cosθ + isinθ) relationship... </p><p>Actually, the key observation: Column 1 + Column 2 = [4, 4, 4]ᵀ which is proportional to Column 3.</p><p>When one column is a linear combination of other columns, the determinant = 0.</p><p><strong>Step 6: Verify the relationship</strong></p><p>4cos²α - 4sin²α = 4(cos²α - sin²α) and 4cos²α + 4sin²α = 4</p><p>Since cos²α + sin²α = 1, we have: Column 1 + Column 2 represents a combination that always yields the constant 4 in each row (matching Column 3).</p><p>Therefore D = 0 for ALL values of α, β, γ.</p><p><strong>∴ Answer: a (independent of α, β, and γ)</strong></p>
Correct Answer: a