Write down the general form of a $2 \times 2$ rotation matrix. Let $R$ be a real $2 \times 2$ matrix with positive determinant such that $|R \mathbf{x}|=|\mathbf{x}|$ for all $\mathbf{x} \in \mathbb{R}^{2}$. Show that $R$ is a rotation matrix.

Let

$$J=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)$$

Show that any real $2 \times 2$ matrix $A$ which satisfies $A J=J A$ can be written as $A=\lambda R$, where $\lambda$ is a real number and $R$ is a rotation matrix.