Problem

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

Solutions relying entirely on calculator technology are not acceptable.

$$M=\begin{pmatrix} 2&0&3\\ 0&4&3\\ 0&4&0 \end{pmatrix}$$

Given that $M$ has exactly two distinct eigenvalues $\lambda_1$ and $\lambda_2$ where $\lambda_1 < \lambda_2$

(a) determine a normalised eigenvector corresponding to the eigenvalue $\lambda_1$

The line $l_1$ has equation

$$\mathbf r=\begin{pmatrix}4\\1\\0\end{pmatrix}+\mu\begin{pmatrix}2\\0\\-1\end{pmatrix},$$

where $\mu$ is a scalar parameter.

The transformation $T$ is represented by $M$.

The line $l_1$ is transformed by $T$ to the line $l_2$

(b) Determine a vector equation for $l_2$, giving your answer in the form $\mathbf r=\mathbf b+\lambda\mathbf c$ where $\mathbf b$ and $\mathbf c$ are constant vectors.