Paper 1, Section I, A

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

(i) Find the values of $\alpha$ for which the inverse map $\Phi^{-1}$ exists, as well as the inverse map itself in these cases.

(ii) When $\boldsymbol{\Phi}$ is not invertible, find its image and kernel. What is the value of the rank and the value of the nullity of $\Phi$ ?

(iii) Let $\mathbf{y}=\mathbf{\Phi}(\mathbf{x})$. 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 $x_{i}=B_{i j} y_{j}$.

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