| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Find P and D for diagonalization / matrix powers |
| Difficulty | Standard +0.3 This is a standard FP3 diagonalization question with a symmetric tridiagonal matrix. Finding eigenvalues requires solving a cubic (which factors nicely), finding eigenvectors is routine, and normalization is straightforward calculation. While it involves multiple steps and 3×3 matrices, it follows a completely standard algorithm with no conceptual challenges or novel insights required—slightly easier than average due to the clean structure. |
| Spec | 4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(\det(\mathbf{A}-\lambda\mathbf{I})=0\) or determinant set to zero | M1 | Either statement sufficient; may be implied by attempt to form characteristic equation |
| \((2-\lambda)[(2-\lambda)^2-2]=0\) or \(\lambda^3 - 6\lambda^2 + 10\lambda - 4 = 0\) | M1 | Recognisable attempt at characteristic equation — sign errors only |
| \(\lambda = 2\) | B1 | \(\lambda = 2\) from any working |
| Solve \(\lambda^2 - 4\lambda + 2 = 0\) | M1 | Attempt to solve quadratic |
| \(\lambda = 2,\ 2+\sqrt{2},\ 2-\sqrt{2}\) (allow awrt 3.41 and 0.586) | A1 | Obtains \(2 \pm \sqrt{2}\) oe e.g. \(\frac{4\pm\sqrt{8}}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| States/uses \(\mathbf{Ax}=\lambda\mathbf{x}\) or \((\mathbf{A}-\lambda\mathbf{I})\mathbf{x}=\mathbf{0}\) for at least one eigenvalue | 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 multiples) | A1 A1 A1 (No ft) | A1: one correct; A1: two correct; A1: all correct (allow awrt 1.41 for \(\sqrt{2}\)) |
| Normalised: \(\pm\begin{pmatrix}\frac{1}{\sqrt{2}}\\0\\-\frac{1}{\sqrt{2}}\end{pmatrix},\ \pm\begin{pmatrix}\frac{1}{2}\\\frac{1}{\sqrt{2}}\\\frac{1}{2}\end{pmatrix},\ \pm\begin{pmatrix}\frac{1}{2}\\-\frac{1}{\sqrt{2}}\\\frac{1}{2}\end{pmatrix}\) | A1 (No ft) | All normalised, correct and exact. Must be seen in (b) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(\mathbf{P} = \begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\\0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\end{pmatrix}\), \(\mathbf{D}=\begin{pmatrix}2&0&0\\0&2+\sqrt{2}&0\\0&0&2-\sqrt{2}\end{pmatrix}\) | B1ft, B1ft | B1ft: one correct ft matrix; B1ft: both correct ft matrices, P consistent with D. Eigenvectors in P must be in same order as eigenvalues in D. Must be clear which matrix is which (NB: B0B1 not possible) |
## Question 3(a):
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $\det(\mathbf{A}-\lambda\mathbf{I})=0$ or determinant set to zero | M1 | Either statement sufficient; may be implied by attempt to form characteristic equation |
| $(2-\lambda)[(2-\lambda)^2-2]=0$ or $\lambda^3 - 6\lambda^2 + 10\lambda - 4 = 0$ | M1 | Recognisable attempt at characteristic equation — sign errors only |
| $\lambda = 2$ | B1 | $\lambda = 2$ from any working |
| Solve $\lambda^2 - 4\lambda + 2 = 0$ | M1 | Attempt to solve quadratic |
| $\lambda = 2,\ 2+\sqrt{2},\ 2-\sqrt{2}$ (allow awrt 3.41 and 0.586) | A1 | Obtains $2 \pm \sqrt{2}$ oe e.g. $\frac{4\pm\sqrt{8}}{2}$ |
## Question 3(b):
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| States/uses $\mathbf{Ax}=\lambda\mathbf{x}$ or $(\mathbf{A}-\lambda\mathbf{I})\mathbf{x}=\mathbf{0}$ for at least one eigenvalue | 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 multiples) | A1 A1 A1 (No ft) | A1: one correct; A1: two correct; A1: all correct (allow awrt 1.41 for $\sqrt{2}$) |
| Normalised: $\pm\begin{pmatrix}\frac{1}{\sqrt{2}}\\0\\-\frac{1}{\sqrt{2}}\end{pmatrix},\ \pm\begin{pmatrix}\frac{1}{2}\\\frac{1}{\sqrt{2}}\\\frac{1}{2}\end{pmatrix},\ \pm\begin{pmatrix}\frac{1}{2}\\-\frac{1}{\sqrt{2}}\\\frac{1}{2}\end{pmatrix}$ | A1 (No ft) | All normalised, correct and exact. Must be seen in (b) |
## Question 3(c):
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $\mathbf{P} = \begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\\0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$, $\mathbf{D}=\begin{pmatrix}2&0&0\\0&2+\sqrt{2}&0\\0&0&2-\sqrt{2}\end{pmatrix}$ | B1ft, B1ft | B1ft: one correct ft matrix; B1ft: both correct ft matrices, **P** consistent with **D**. Eigenvectors in **P** must be in same order as eigenvalues in **D**. Must be clear which matrix is which (NB: B0B1 not possible) |
---3.
$$\mathbf { A } = \left( \begin{array} { l l l }
2 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find the eigenvalues of $\mathbf { A }$.
\item Find a normalised eigenvector for each of the eigenvalues of $\mathbf { A }$.
\item Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2015 Q3 [12]}}