(a) In the standard basis of $\mathbb{R}^{2}$, write down the matrix for a rotation through an angle $\theta$ about the origin.

(b) Let $A$ be a real $3 \times 3$ matrix such that $\operatorname{det} A=1$ and $A A^{\mathrm{T}}=I$, where $A^{\mathrm{T}}$ is the transpose of $A$.

(i) Suppose that $A$ has an eigenvector $\mathbf{v}$ with eigenvalue 1 . Show that $A$ is a rotation through an angle $\theta$ about the line through the origin in the direction of $\mathbf{v}$, where $\cos \theta=\frac{1}{2}($ trace $A-1)$.

(ii) Show that $A$ must have an eigenvector $\mathbf{v}$ with eigenvalue 1 .