General expression for exponentiating a matrix? $A= \begin{pmatrix}
        \cos(x) & \sin(x) \\
        -\sin(x) & \cos(x) \\
        \end{pmatrix}$
I found that 
$A^2 = \begin{pmatrix}
        \cos^2(x)-\sin^2(x) & 2\cos(x)\sin(x) \\
        -2\cos(x)\sin(x) & \cos^2(x)-\sin^2(x) \\
        \end{pmatrix}$
$A^2 = \begin{pmatrix}
        \cos(2x) & \sin(2x) \\
        -\sin(2x) & \cos(2x) \\
        \end{pmatrix}$
and
$A^3 = \begin{pmatrix}
        \cos^2(x)[\cos^2(x)-3\sin^2(x)] & \sin(x)[3\cos^2(x)\sin^2(x)] \\
        -\sin(x)[3\cos^2(x)+\sin^2(x)] & \cos(x)[-3\sin^2(x)+\cos^2(x)] \\
        \end{pmatrix}$
$A^3 = \begin{pmatrix}
        \cos(2x)cos(x) - \sin(x)\sin(2x) & \sin(x)\cos(2x) + \cos(x)\sin(2x) \\
        -\cos(x)sin(2x)-\sin(x)\cos(2x) &  \cos(x) + \cos(2x) -\sin(x)\sin(2x) \\
        \end{pmatrix}$
but I am unable to find a general solution. I probably need to perform some algebraic manipulations or whatnot.

In general, following formula holds: (Proof of this formula uses induction.)
$$A^n =\begin{pmatrix}
\cos (nx) & -\sin(nx) \\
\sin(nx) & \cos(nx) 
\end{pmatrix}$$
In fact, matrix $\begin{pmatrix}
\cos (x) & -\sin(x) \\
\sin(x) & \cos(x) 
\end{pmatrix}$ behave similar to $\cos x + i \sin x$.
