# Question 1 (Eigenvalues/Eigenvectors):

## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to expand determinant to find characteristic polynomial | M1 | Other expansion methods possible |
| Correct expansion (need not be simplified, need not equal zero) | A1 | Allow recovery of missing brackets if indicated by later working |
| Attempts to take out factor of $(\lambda - 2)$ | M1 | May first expand to cubic or spot factor |
| $(6-\lambda)\big((3-\lambda)^2-1\big)+2\big(2(\lambda-3)+2\big)+2\big(2-2(3-\lambda)\big)=(6-\lambda)(4-\lambda)(2-\lambda)+4(\lambda-2)+4(\lambda-2)$ | | |
| $=(\lambda-2)\big((6-\lambda)(4-\lambda)+4+4\big)$ | | |
| Obtains correct factor of $(\lambda-2)^2$ and deduces 2 is a repeated eigenvalue | A1* | Must see **statement** about 2 being repeated; just listing 2 twice insufficient |
| Identifies 8 as other eigenvalue | B1 | B0 if not identified in (a) but full marks can be scored in (b) and (c) |

## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses correct method to find at least one eigenvector | M1 | |
| Obtains one correct eigenvector for $\lambda=8$ | A1 | |
| Obtains one correct eigenvector for $\lambda=2$ | A1 | |
| Obtains two correct linearly independent eigenvectors for $\lambda=2$ | A1 | Common examples: $\begin{pmatrix}1\\1\\-1\end{pmatrix}, \begin{pmatrix}1\\3\\1\end{pmatrix}, \begin{pmatrix}2\\5\\1\end{pmatrix}, \begin{pmatrix}1\\4\\2\end{pmatrix}$ |

## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Forms matrix with three **different non-zero** eigenvectors as columns (or normalised/scaled versions) | M1 | |
| Correct matrix with eigenvectors as columns in any order | A1ft | Follow through their three **different** vectors which are **not multiples** of any other |

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