The map $\Phi(\mathbf{x})=\mathbf{n} \times(\mathbf{x} \times \mathbf{n})+\alpha(\mathbf{n} \cdot \mathbf{x}) \mathbf{n}$ is defined for $\mathbf{x} \in \mathbb{R}^{3}$, where $\mathbf{n}$ is a unit vector in $\mathbb{R}^{3}$ and $\alpha$ is a constant.

(a) Find the inverse map $\Phi^{-1}$, when it exists, and determine the values of $\alpha$ for which it does.

(b) When $\Phi$ is not invertible, find its image and kernel, and explain geometrically why these subspaces are perpendicular.

(c) Let $\mathbf{y}=\Phi(\mathbf{x})$. Find the components $A_{i j}$ of the matrix $A$ such that $y_{i}=A_{i j} x_{j}$. When $\Phi$ is invertible, find the components of the matrix $B$ such that $x_{i}=B_{i j} y_{j}$.

(d) Now let $A$ be as defined in (c) for the case $\mathbf{n}=\frac{1}{\sqrt{3}}(1,1,1)$, and let

$$C=\frac{1}{3}\left(\begin{array}{rrr} 2 & 2 & -1 \\ -1 & 2 & 2 \\ 2 & -1 & 2 \end{array}\right)$$

By analysing a suitable determinant, for all values of $\alpha$ find all vectors $\mathbf{x}$ such that $A \mathbf{x}=C \mathbf{x}$. Explain your results by interpreting $A$ and $C$ geometrically.