Question

If $A(x)=\begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix}$, then $(A(x))^{-1}$ is equal to :

• A. $A(x)$
• B. $-A(x)$
• C. $A(-x)$
• D. None

Answer 1

Answer: (C)

$A(-x)$

Answer 2

Key Idea: $A(x)$ is a rotation matrix, so it is orthogonal and satisfies $A(x)^{-1}=A(x)^{T}$.
Computation:

$$A(x)^{T} =\begin{bmatrix} \cos x & \sin x\\ -\sin x & \cos x \end{bmatrix} =\begin{bmatrix} \cos(-x) & -\sin(-x)\\ \sin(-x) & \cos(-x) \end{bmatrix} = A(-x).$$

Hence, $(A(x))^{-1}=A(-x)$.