Let $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then
Step-by-Step Solution
Key Concept: The inverse of the matrix function $F(x)$ is identified as $F(-x)$ by demonstrating that their product $F(x)F(-x)$ equals the identity matrix $I$.
<div>We have, $F(x)F(-x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos x & \sin x & 0 \\ -\sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I_3$<br/>$\Rightarrow F(-x)$ is the inverse of matrix $F(x)$ i.e. $[F(x)]^{-1} = F(-x)$.</div>
Correct Answer: D