Question

Let $$ G(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ If $ x + y = 0 $, then evaluate $ G(x)G(y) $

• A. Null Matrix
• B. Skew Symmetric Matrix
• C. Identity Matrix
• D. Symmetric Matrix

Hint:

If \( G(x) \) is a rotation matrix, then \( G(x)G(y) = G(x+y) \). Use this property when combining rotation matrices.

Answer 1

The Correct Option is C

We are given: \[ G(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \] This is a rotation matrix about the z-axis in 3D. The composition of two such rotations is equivalent to rotation by the sum of the angles. That is: \[ G(x)G(y) = G(x + y) \] So, if \( x + y = 0 \), then: \[ G(x)G(y) = G(0) \] Now compute \( G(0) \): \[ G(0) = \begin{bmatrix} \cos 0 & -\sin 0 & 0 \\ \sin 0 & \cos 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I_3 \] Hence, \( G(x)G(y) = I \), the identity matrix.