= Solution
A real square matrix $M$ is an <orthogonal matrix> when $M^TM=I$. The planar rotation
$$
R=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}
$$
obeys $R^TR=I$ by the <Pythagorean trigonometric identity>.
For $A=\begin{pmatrix}a&b\\b&c\end{pmatrix}$, the off-diagonal entry of $RAR^T$ is $\frac12(a-c)\sin2\theta+b\cos2\theta$. Thus
$$
\boxed{(a-c)\sin2\theta+2b\cos2\theta=0.}
$$
If $(a-c,b)\ne(0,0)$, equivalently $\theta=\frac12\operatorname{atan2}(-2b,a-c)+k\pi/2$; if $A$ is scalar, every angle works.
Because $RAR^T$ is an <orthogonal similarity>, its diagonal entries are the eigenvalues
$$
\lambda_\pm=\frac{\operatorname{tr}A\pm\sqrt{(\operatorname{tr}A)^2-4\det A}}2.
$$
The matrix $C=\begin{pmatrix}1&2\\2&1\end{pmatrix}$ has eigenvalues $3,-1$ along $(1,1)^T,(1,-1)^T$. Raising its orthogonal diagonalization to the even power gives
$$
\boxed{C^{2N}=\frac12\begin{pmatrix}3^{2N}+1&3^{2N}-1\\3^{2N}-1&3^{2N}+1\end{pmatrix}.}
$$
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