Problem

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The matrix $M$ is given by

$$M = \begin{pmatrix} -2 & 2 & 2 \\ -3 & -7 & -3 \\ 1 & 1 & -3 \end{pmatrix}$$

(a) Show that $\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$ is an eigenvector of $M$, stating the corresponding eigenvalue.

(b) Show that the eigenvalue from part (a) is the only eigenvalue of M.

The transformation $M : \mathbb{R}^3 \to \mathbb{R}^3$ is represented by the matrix $M$.

(c) Determine a Cartesian equation for the image under M of the line with equation

$$\frac{x-1}{1} = \frac{y-1}{2} = \frac{z}{-3}$$