(a) Let $A$ and $A^{\prime}$ be the matrices of a linear map $L$ on $\mathbb{C}^{2}$ relative to bases $\mathcal{B}$ and $\mathcal{B}^{\prime}$ respectively. In this question you may assume without proof that $A$ and $A^{\prime}$ are similar.

(i) State how the matrix $A$ of $L$ relative to the basis $\mathcal{B}=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}$ is constructed from $L$ and $\mathcal{B}$. Also state how $A$ may be used to compute $L \mathbf{v}$ for any $\mathbf{v} \in \mathbb{C}^{2}$.

(ii) Show that $A$ and $A^{\prime}$ have the same characteristic equation.

(iii) Show that for any $k \neq 0$ the matrices

$$\left(\begin{array}{ll} a & c \\ b & d \end{array}\right) \text { and }\left(\begin{array}{cc} a & c / k \\ b k & d \end{array}\right)$$

are similar. [Hint: if $\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}$ is a basis then so is $\left\{k \mathbf{e}_{1}, \mathbf{e}_{2}\right\}$.]

(b) Using the results of (a), or otherwise, prove that any $2 \times 2$ complex matrix $M$ with equal eigenvalues is similar to one of

$$\left(\begin{array}{ll} a & 0 \\ 0 & a \end{array}\right) \text { and }\left(\begin{array}{ll} a & 1 \\ 0 & a \end{array}\right) \text { with } a \in \mathbb{C} .$$

(c) Consider the matrix

$$B(r)=\frac{1}{2}\left(\begin{array}{ccc} 1+r & 1-r & 1 \\ 1-r & 1+r & -1 \\ -1 & 1 & 2 r \end{array}\right)$$

Show that there is a real value $r_{0}>0$ such that $B\left(r_{0}\right)$ is an orthogonal matrix. Show that $B\left(r_{0}\right)$ is a rotation and find the axis and angle of the rotation.