Considering the rotation matrix:
$$
A(\theta) = \left( \begin{array}{cc}
\cos\space\theta & -\sin \space\theta \\
\sin \space\theta & \cos\space\theta \\
\end{array} \right)
$$
How can I calculate $(A(\theta))^n$ where n ≥ 1 ? I'm not sure what to do nor how to write it.
Best Answer
Notice that $[A(\theta)]^n = A(n\theta)$, based on geometric interpretation of rotations. If you apply $A(\theta)$ twice, it is the same as rotating by $\theta$ twice, or $2\theta$. The same is true for an arbitrary number of rotations. That shows you have to write $A(\theta)^n$, namely as the same matrix with $\theta$ replaced by $n\theta$.