Originally Posted by

**hatsoff** This exercise has been troubling me. It seems like it should be simple, but I can't seem to crack it.

Let $\displaystyle A=\left[\begin{array}{cc} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta}\end{array} \right]$

Find $\displaystyle A^k$, where $\displaystyle k\in\mathbb{N}$

To solve this, I tried to find a pattern in the following:

$\displaystyle A^2=\left[\begin{array}{cc} \cos^2{\theta}-\sin^2{\theta} & 2\sin{\theta}\cos{\theta} \\ -2\sin{\theta}\cos{\theta} & \cos^2{\theta}-\sin^2{\theta}\end{array} \right]$

$\displaystyle A^3=\left[\begin{array}{cc} \cos^3{\theta}-3\sin^2{\theta}\cos{\theta} & 3\sin{\theta}\cos^2{\theta}-\sin^3{\theta} \\ -3\sin{\theta}\cos^2{\theta}+\sin^3{\theta} & \cos^3{\theta}-3\sin^2{\theta}\cos{\theta}\end{array} \right]$

$\displaystyle A^4=\left[\begin{array}{cc}\cos^4{\theta}-6\sin^2{\theta}\cos^2{\theta}+\sin^4{\theta}&4\sin {\theta}\cos^3{\theta}-4\sin^3{\theta}\cos{\theta}\\-4\sin{\theta}\cos^3{\theta}+4\sin^3{\theta}\cos{\t heta}&\cos^4{\theta}-6\sin^2{\theta}\cos^2{\theta}+\sin^4{\theta}\end{a rray}\right]$

From this I observe that

$\displaystyle A^k=\frac{1}{k^2}\left[\begin{array}{cc}kf'(\theta)&k^2f(\theta)\\f''(\th eta)&kf'(\theta)\end{array}\right]$

But I can't seem to generalize $\displaystyle f(\theta)$ in terms of $\displaystyle k$. Any ideas?