<p>If <span>\(A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}\)</span> and <span>\(A + A^T = I\)</span>, find the value of <span>\(\alpha\)</span>.</p>
Step-by-Step Solution
Key Concept: Recognize that A is a rotation matrix, and use the condition A + A^T = I to derive a trigonometric equation. The transpose of a rotation matrix equals its inverse, so A^T = [cos α, sin α; -sin α, cos α], allowing you to sum and equate to I.
<p><strong>Step 1:</strong> Write the transpose of A.</p><p>A^T = [cos α, sin α; -sin α, cos α]</p><p><strong>Step 2:</strong> Add A + A^T and set equal to I.</p><p>A + A^T = [cos α + cos α, -sin α + sin α; sin α - sin α, cos α + cos α] = [2cos α, 0; 0, 2cos α]</p><p><strong>Step 3:</strong> Equate A + A^T = I.</p><p>[2cos α, 0; 0, 2cos α] = [1, 0; 0, 1]</p><p>This gives: 2cos α = 1</p><p><strong>Step 4:</strong> Solve for α.</p><p>cos α = 1/2</p><p>α = 2nπ ± π/3, where n ∈ ℤ</p><p><strong>Verification:</strong> When α = π/3: cos(π/3) = 1/2 ✓, and A + A^T equals I ✓</p><p>∴ Answer: α = 2nπ ± π/3, n ∈ ℤ</p>
Correct Answer: \alpha = 2n\pi \pm \dfrac{\pi}{3}, n \in \mathbb{Z}