A2 June 2019 Q2
2. The matrix \(\mathbf{A}\) is given by
\[\mathbf{A} = \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix}\]| Scheme | Marks | AO |
|---|---|---|
| \(|\mathbf{A} - \lambda\mathbf{I}| = \begin{vmatrix} 6 - \lambda & -2 & 2 \\ -2 & 3 - \lambda & -1 \\ 2 & -1 & 3 - \lambda \end{vmatrix} = (6 - \lambda)[\ldots] - (-2)[\ldots] + 2[\ldots] = \ldots\) | M1 | 1.1b |
| \((6 - \lambda)\left((3 - \lambda)^2 - 1\right) + 2\big(2(\lambda - 3) + 2\big) + 2\big(2 - 2(3 - \lambda)\big)\ (= 0)\) \(\left(\lambda^3 - 12\lambda^2 + 36\lambda - 32 = 0\right)\) | A1 | 1.1b |
| \(= (\lambda - 2)(\lambda^2 + \ldots\lambda + \ldots)\) | M1 | 2.1 |
| \(= (\lambda - 2)(\lambda^2 - 10\lambda + 16) = (\lambda - 2)^2(\lambda - 8) \Rightarrow \lambda = 2\) is a repeated eigenvalue * | A1* | 2.2a |
| \(\lambda = 8\) | B1 | 1.1b |
| (5) |
Notes
M1: Attempts to expand the determinant to find the characteristic polynomial.
Note: other methods of expanding the determinant are possible. If unsure send to review.
A1: Correct expansion need not be simplified. (Need not see set equal to zero) Allow recovery of missing brackets if indicated by later working.
M1: Attempts to take out a factor of \((\lambda - 2)\) of their equation (may first expand to cubic or may spot the factor and take out without full expansion). E.g
\((6 - \lambda)\left((3 - \lambda)^2 - 1\right) + 2\big(2(\lambda - 3) + 2\big) + 2\big(2 - 2(3 - \lambda)\big) = (6 - \lambda)(4 - \lambda)(2 - \lambda) + 4(\lambda - 2) + 4(\lambda - 2)\)
\(= (\lambda - 2)\big({-(6 - \lambda)(4 - \lambda)} + 4 + 4\big)\)
(corrected from the printed mark scheme: the last line is printed as \((\lambda - 2)\big((6 - \lambda)(4 - \lambda) + 4 + 4\big)\), without the minus sign)
This is for a method that will allow \(\lambda\) to be shown as a repeated eigenvalue, so just stating two solutions is not sufficient, factorisation must be seen.
A1*: Obtains a correct factor of \((\lambda - 2)^2\) and deduces that 2 is a repeated eigenvalue. Must see statement about 2 being repeated. (Just listing 2 twice is not sufficient.)
B1: (Note this is A1 on ePEN) Obtains and identifies 8 as the other eigenvalue (B0 if not identified in (a) but full marks can be scored in (b) and (c) for use of 8 as eigenvalue)
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix}\mathbf{v} = 2\begin{pmatrix} x \\ y \\ z \end{pmatrix}\) or \(\begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix}\mathbf{v} = 8\begin{pmatrix} x \\ y \\ z \end{pmatrix} \Rightarrow \mathbf{v} = \ldots\) | M1 | 1.1b |
| Obtains any multiple of \(\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\) for \(\lambda = 8\) | A1 | 1.1b |
| Obtains any (non-zero) multiple or linear combination of \(\begin{pmatrix} -1 \\ 0 \\ 2 \end{pmatrix}\text{ or }\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}\text{ or }\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\) for \(\lambda = 2\) | A1 | 1.1b |
| Obtains a different linear combination or (non-zero) multiple of different vector from \(\begin{pmatrix} -1 \\ 0 \\ 2 \end{pmatrix}\text{ or }\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}\text{ or }\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\) for \(\lambda = 2\) | A1 | 3.1a |
| (4) |
Notes
M1: Uses a correct method to find at least one eigenvector
A1: Obtains one correct eigenvector for \(\lambda = 8\)
A1: Obtains one correct eigenvector for \(\lambda = 2\)
A1: Obtains two correct linearly independent eigenvectors for \(\lambda = 2\)
Note some other common eigenvectors for \(\lambda = 2\) are \(\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}, \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix}\)
| Scheme | Marks | AO |
|---|---|---|
| Forms a matrix with their eigenvectors as columns | M1 | 1.2 |
| E.g. \(\begin{pmatrix} -1 & 1 & 2 \\ 0 & 2 & -1 \\ 2 & 0 & 1 \end{pmatrix}\) | A1ft | 1.1b |
| (2) | ||
| (11 marks) |
Notes
M1: Forms a matrix with their three different non-zero eigenvectors as columns or with their normalised (or any scaled version) of their eigenvectors.
A1ft: Correct matrix with the eigenvectors (normalised/scaled) as columns in any order (follow through their three different vectors which are not multiples of any other)