| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Reconstruct matrix from eigenvalues and eigenvectors |
| Difficulty | Standard +0.8 This is a standard FP3 diagonalization question requiring normalization of eigenvectors to form an orthogonal matrix, verification of the orthogonality property, and reconstruction of M via similarity transformation. While it involves multiple steps and matrix manipulation, it follows a well-established procedure taught explicitly in Further Maths with no novel insight required. The computational work is moderate but routine for this level. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\mathbf{P} = \begin{pmatrix} \frac{2}{3} & -\frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & \frac{1}{3} & -\frac{2}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2 \end{pmatrix}\) | M1 | Attempt unit eigenvectors |
| Correct matrix | A1 | |
| \(\mathbf{D} = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{pmatrix}\) | M1 | Correct form for \(\mathbf{D}\) with eigenvalues on diagonal |
| Consistent with \(\mathbf{P}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\mathbf{MP} = \mathbf{PD}\) or \(\mathbf{P}^{-1}\mathbf{M} = \mathbf{DP}^{-1}\) | M1 | |
| \(\mathbf{M} = \mathbf{PDP}^{-1}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\mathbf{P}^{-1} = \mathbf{P}^T = \frac{1}{3}\begin{pmatrix} 2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2 \end{pmatrix}\) | B1ft | Correct matrix; allow transpose of their \(\mathbf{P}\) |
| \(\mathbf{PD} = \frac{1}{3}\begin{pmatrix} 2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2 \end{pmatrix}\begin{pmatrix} 5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 10 & -4 & -1 \\ 10 & 2 & 2 \\ 5 & 4 & -2 \end{pmatrix}\) or \(\mathbf{DP}^{-1} = \frac{1}{3}\begin{pmatrix} 10 & 10 & 5 \\ -4 & 2 & 4 \\ -1 & 2 & -2 \end{pmatrix}\) | M1A1 | M1: Attempt \(\mathbf{PD}\) or \(\mathbf{DP}^{-1}\) where \(\mathbf{D} \neq k\mathbf{I}\); A1: Correct matrix |
| \(\mathbf{M} = \frac{1}{9}\begin{pmatrix} 10 & -4 & -1 \\ 10 & 2 & 2 \\ 5 & 4 & -2 \end{pmatrix}\begin{pmatrix} 2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2 \end{pmatrix} = \begin{pmatrix} 3 & 2 & 0 \\ 2 & 2 & 2 \\ 0 & 2 & 1 \end{pmatrix}\) | M1A1 | M1: Completes correctly to find \(\mathbf{M}\); A1: Correct \(\mathbf{M}\) |
# Question 6:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{P} = \begin{pmatrix} \frac{2}{3} & -\frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & \frac{1}{3} & -\frac{2}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2 \end{pmatrix}$ | M1 | Attempt unit eigenvectors |
| Correct matrix | A1 | |
| $\mathbf{D} = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{pmatrix}$ | M1 | Correct form for $\mathbf{D}$ with eigenvalues on diagonal |
| Consistent with $\mathbf{P}$ | A1 | |
**(4 marks)**
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{MP} = \mathbf{PD}$ or $\mathbf{P}^{-1}\mathbf{M} = \mathbf{DP}^{-1}$ | M1 | |
| $\mathbf{M} = \mathbf{PDP}^{-1}$ | A1 | |
**(2 marks)**
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{P}^{-1} = \mathbf{P}^T = \frac{1}{3}\begin{pmatrix} 2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2 \end{pmatrix}$ | B1ft | Correct matrix; allow transpose of their $\mathbf{P}$ |
| $\mathbf{PD} = \frac{1}{3}\begin{pmatrix} 2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2 \end{pmatrix}\begin{pmatrix} 5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 10 & -4 & -1 \\ 10 & 2 & 2 \\ 5 & 4 & -2 \end{pmatrix}$ **or** $\mathbf{DP}^{-1} = \frac{1}{3}\begin{pmatrix} 10 & 10 & 5 \\ -4 & 2 & 4 \\ -1 & 2 & -2 \end{pmatrix}$ | M1A1 | M1: Attempt $\mathbf{PD}$ or $\mathbf{DP}^{-1}$ where $\mathbf{D} \neq k\mathbf{I}$; A1: Correct matrix |
| $\mathbf{M} = \frac{1}{9}\begin{pmatrix} 10 & -4 & -1 \\ 10 & 2 & 2 \\ 5 & 4 & -2 \end{pmatrix}\begin{pmatrix} 2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2 \end{pmatrix} = \begin{pmatrix} 3 & 2 & 0 \\ 2 & 2 & 2 \\ 0 & 2 & 1 \end{pmatrix}$ | M1A1 | M1: Completes correctly to find $\mathbf{M}$; A1: Correct $\mathbf{M}$ |
> **Note:** Failure to use unit eigenvectors in (a) could score M0A0M1A1 in (a) and B1ftM1A0M1A0 in (c)
**(5 marks) — Total: 11**
---6. The symmetric matrix $\mathbf { M }$ has eigenvectors $\left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } - 2 \\ 1 \\ 2 \end{array} \right)$ and $\left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)$ with eigenvalues 5, 2 and - 1 respectively.
\begin{enumerate}[label=(\alph*)]
\item Find an orthogonal matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that
$$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$
Given that $\mathbf { P } ^ { - 1 } = \mathbf { P } ^ { \mathrm { T } }$
\item show that
$$\mathbf { M } = \mathbf { P D P } ^ { - 1 }$$
\item Hence find the matrix $\mathbf { M }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2014 Q6 [11]}}