Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 1 / 2A (Vectors and Matrices)

The vector triple product identity gives $\mathbf{n}\times (\mathbf{n}\times \mathbf{x})=\mathbf{n}(\mathbf{n}\cdot \mathbf{x})-\mathbf{x}$, and henceThus $\Phi$ is the identity on the orthogonal complement ${\mathbf{n}}^{\perp }$ and multiplies the component parallel to $\mathbf{n}$ by $\alpha$. It is invertible exactly when $\alpha \ne 0$, and then

The only noninvertible case is $\alpha =0$. Then $\Phi$ is the orthogonal projection onto ${\mathbf{n}}^{\perp }$, soThese dimensions agree with the rank-nullity theorem.

In Einstein notation, $(\mathbf{n}\cdot \mathbf{x})n_i=n_i{n}_j{x}_j$, so the matrix and, for $\alpha \ne 0$, its matrix inverse are