Matrices & Determinants
Rotation and Orthogonal Matrices
Grade None
Question:
<p>Let \(A_\alpha = \begin{bmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}\), then:</p>
<p>(a) \(A_{\alpha+\beta} = A_\alpha A_\beta\)</p>
<p>(b) \(A_\alpha^{-1} = A_{-\alpha}\)</p>
<p>(c) \(A_\alpha^{-1} = -A_\alpha\)</p>
<p>(d) \(A_\alpha^2 = -I\)</p>
Step-by-Step Solution
Key Concept: Recognize rotation matrices and their composition properties under matrix multiplication.
<p>$A_\alpha$ is a rotation matrix about the z-axis by angle $\alpha$.</p><p>(a) Composition of rotations: $A_{\alpha+\beta} = A_\alpha A_\beta$ ✓</p><p>(b) Inverse rotation: $A_\alpha^{-1} = A_{-\alpha}$ ✓</p><p>(c) False: $A_\alpha^{-1} = A_{-\alpha} \neq -A_\alpha$</p><p>(d) False: $A_\alpha^2 = A_{2\alpha} \neq -I$</p>
Correct Answer: a, b