FP3 June 2015 Q3
3. \[\mathbf{A} = \begin{pmatrix}2 & 1 & 0\\ 1 & 2 & 1\\ 0 & 1 & 2\end{pmatrix}\]
| Scheme | Marks |
|---|---|
| \(\det(\mathbf{A} - \lambda\mathbf{I}) = 0\) or \(\begin{vmatrix}2-\lambda & 1 & 0\\ 1 & 2-\lambda & 1\\ 0 & 1 & 2-\lambda\end{vmatrix} = 0\) | M1 |
| \((2 - \lambda)((2 - \lambda)^2 - 1) - (2 - \lambda) = 0\) or \((2 - \lambda)\left[(2 - \lambda)^2 - 2\right] = 0\) \(\left(\lambda^3 - 6\lambda^2 + 10\lambda - 4 = 0\right)\) | M1 |
| \((2 - \lambda)((\lambda^2 - 4\lambda + 2)) = 0\) | |
| \(\lambda = 2,\ 2 + \sqrt{2},\ 2 - \sqrt{2}\) Allow awrt 3.41 and 0.586 | B1M1A1 |
| (5) |
Notes
M1: Either statement is sufficient. May also be implied by an attempt to form the characteristic equation
M1: Recognisable attempt at characteristic equation – sign errors only.
B1: \(\lambda = 2\) from any working
M1: Attempt to solve (usual rules) \(\lambda^2 - 4\lambda + 2 = 0\)
A1: Obtains \(2 \pm \sqrt{2}\) oe e.g. \(\frac{4 \pm \sqrt{8}}{2}\)
| Scheme | Marks |
|---|---|
| \(\begin{pmatrix}2 & 1 & 0\\ 1 & 2 & 1\\ 0 & 1 & 2\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix} = 2\begin{pmatrix}x\\ y\\ z\end{pmatrix}\) or \((2 + \sqrt{2})\begin{pmatrix}x\\ y\\ z\end{pmatrix}\) or \((2 - \sqrt{2})\begin{pmatrix}x\\ y\\ z\end{pmatrix}\) States or uses \(\mathbf{A}\boldsymbol{x} = \lambda\boldsymbol{x}\) or \((\mathbf{A} - \lambda\mathbf{I})\boldsymbol{x} = \mathbf{0}\) for at least one of their eigenvalues | M1 |
| \(\begin{pmatrix}1\\ 0\\ -1\end{pmatrix},\ \begin{pmatrix}1\\ \sqrt{2}\\ 1\end{pmatrix},\ \begin{pmatrix}1\\ -\sqrt{2}\\ 1\end{pmatrix}\) (any multiple of these) | A1 A1 A1 No ft here |
| \(\pm\begin{pmatrix}\frac{1}{\sqrt{2}}\\ 0\\ -\frac{1}{\sqrt{2}}\end{pmatrix},\ \pm\begin{pmatrix}\frac{1}{2}\\ \frac{1}{\sqrt{2}}\\ \frac{1}{2}\end{pmatrix},\ \pm\begin{pmatrix}\frac{1}{2}\\ -\frac{1}{\sqrt{2}}\\ \frac{1}{2}\end{pmatrix}\) | A1 No ft here |
| (5) |
Notes
A1: One correct eigenvector (allow awrt 1.41 for \(\sqrt{2}\))
A1: Two correct eigenvectors (allow awrt 1.41 for \(\sqrt{2}\))
A1: All eigenvectors correct (allow awrt 1.41 for \(\sqrt{2}\))
A1: All normalised and correct and exact. Allow equivalent forms e.g. \(\dfrac{1}{\sqrt{2}}\begin{pmatrix}1\\ 0\\ -1\end{pmatrix}\) (Must be seen in (b))
| Scheme | Marks |
|---|---|
| \(\mathbf{P} = \begin{pmatrix}\frac{1}{\sqrt{2}} & \frac{1}{2} & \frac{1}{2}\\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}} & \frac{1}{2} & \frac{1}{2}\end{pmatrix}\quad \mathbf{D} = \begin{pmatrix}2 & 0 & 0\\ 0 & 2 + \sqrt{2} & 0\\ 0 & 0 & 2 - \sqrt{2}\end{pmatrix}\) | B1ft, B1ft |
| (2) | |
| (12 marks) |
Notes
B1ft: One correct ft matrix. If awarding for \(\mathbf{P}\) they must be using their normalised vectors
B1ft: Both correct ft matrices and \(\mathbf{P}\) consistent with \(\mathbf{D}\). The eigenvectors in \(\mathbf{P}\) must be in the same order as the eigenvalues in \(\mathbf{D}\).
For both B marks it must be clear or implied which matrix is which. (NB: B0B1 is not possible)