(a) Consider the matrix

$$A=\left(\begin{array}{rrr} \mu & 1 & 1 \\ 2 & -\mu & 0 \\ -\mu & 2 & 1 \end{array}\right)$$

Find the kernel of $A$ for each real value of the constant $\mu$. Hence find how many solutions $\mathbf{x} \in \mathbb{R}^{3}$ there are to

$$A \mathbf{x}=\left(\begin{array}{l} 1 \\ 1 \\ 2 \end{array}\right)$$

depending on the value of $\mu$. [There is no need to find expressions for the solution(s).]

(b) Consider the reflection map $\Phi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ defined as

$$\Phi: \mathbf{x} \mapsto \mathbf{x}-2(\mathbf{x} \cdot \mathbf{n}) \mathbf{n}$$

where $\mathbf{n}$ is a unit vector normal to the plane of reflection.

(i) Find the matrix $H$ which corresponds to the map $\Phi$ in terms of the components of $\mathbf{n}$.

(ii) Prove that a reflection in a plane with unit normal $\mathbf{n}$ followed by a reflection in a plane with unit normal vector $\mathbf{m}$ (both containing the origin) is equivalent to a rotation along the line of intersection of the planes with an angle twice that between the planes.

[Hint: Choose your coordinate axes carefully.]

(iii) Briefly explain why a rotation followed by a reflection or vice-versa can never be equivalent to another rotation.

Part IA, 2021 List of Questions