What is the definition of an orthogonal matrix $M$ ?

Write down a $2 \times 2$ matrix $R$ representing the rotation of a 2-dimensional vector $(x, y)$ by an angle $\theta$ around the origin. Show that $R$ is indeed orthogonal.

Take a matrix

$$A=\left(\begin{array}{ll} a & b \\ b & c \end{array}\right)$$

where $a, b, c$ are real. Suppose that the $2 \times 2$ matrix $B=R A R^{T}$ is diagonal. Determine all possible values of $\theta$.

Show that the diagonal entries of $B$ are the eigenvalues of $A$ and express them in terms of the determinant and trace of $A$.

Using the above results, or otherwise, find the elements of the matrix

$$\left(\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right)^{2 N}$$

as a function of $N$, where $N$ is a natural number.