| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find P and D for A = PDP⁻¹ |
| Difficulty | Challenging +1.2 This is a standard diagonalization problem from FP2 involving a repeated eigenvalue. While it requires finding eigenvalues via characteristic equation, finding eigenvectors for a repeated eigenvalue (which can be tricky), and constructing P and D, these are well-practiced techniques. The question guides students through each step explicitly, and the 3×3 matrix with nice integer entries keeps calculations manageable. Slightly above average difficulty due to the repeated eigenvalue complication, but still a textbook exercise. |
| Spec | 4.03l Singular/non-singular matrices4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \( | \mathbf{A} - \lambda\mathbf{I} | = \begin{vmatrix} 6-\lambda & -2 & 2 \\ -2 & 3-\lambda & -1 \\ 2 & -1 & 3-\lambda \end{vmatrix} = (6-\lambda)[\ldots] - (-2)[\ldots] + 2[\ldots] = \ldots\) |
| \((6-\lambda)((3-\lambda)^2 - 1) + 2(2(\lambda-3)+2) + 2(2-2(3-\lambda)) = 0\) giving \(\lambda^3 - 12\lambda^2 + 36\lambda - 32 = 0\) | A1 | Correct characteristic equation |
| \(= (\lambda - 2)(\lambda^2 + \ldots\lambda + \ldots)\) | M1 | Attempts factorisation |
| \(= (\lambda-2)(\lambda^2 - 10\lambda + 16) = (\lambda-2)^2(\lambda-8) \Rightarrow \lambda = 2\) is a repeated eigenvalue | A1* | Fully correct factorisation with conclusion |
| \(\lambda = 8\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Sets up \((A - \lambda I)\mathbf{v} = \mathbf{0}\) for \(\lambda = 2\) or \(\lambda = 8\) | M1 | |
| Any multiple of \(\begin{pmatrix}2\\-1\\1\end{pmatrix}\) for \(\lambda = 8\) | A1 | |
| Any non-zero multiple or linear combination of \(\begin{pmatrix}-1\\0\\2\end{pmatrix}\) or \(\begin{pmatrix}1\\2\\0\end{pmatrix}\) or \(\begin{pmatrix}0\\1\\1\end{pmatrix}\) for \(\lambda = 2\) | A1 | |
| Obtains a different linear combination or non-zero multiple of a different vector from the above options for \(\lambda = 2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Forms a matrix with their eigenvectors as columns | M1 | |
| E.g. \(\begin{pmatrix}-1 & 1 & 2\\ 0 & 2 & -1\\ 2 & 0 & 1\end{pmatrix}\) | A1ft |
# Question 2:
## Part (a)
| Working | Mark | Guidance |
|---------|------|----------|
| $|\mathbf{A} - \lambda\mathbf{I}| = \begin{vmatrix} 6-\lambda & -2 & 2 \\ -2 & 3-\lambda & -1 \\ 2 & -1 & 3-\lambda \end{vmatrix} = (6-\lambda)[\ldots] - (-2)[\ldots] + 2[\ldots] = \ldots$ | M1 | Attempt at determinant expansion |
| $(6-\lambda)((3-\lambda)^2 - 1) + 2(2(\lambda-3)+2) + 2(2-2(3-\lambda)) = 0$ giving $\lambda^3 - 12\lambda^2 + 36\lambda - 32 = 0$ | A1 | Correct characteristic equation |
| $= (\lambda - 2)(\lambda^2 + \ldots\lambda + \ldots)$ | M1 | Attempts factorisation |
| $= (\lambda-2)(\lambda^2 - 10\lambda + 16) = (\lambda-2)^2(\lambda-8) \Rightarrow \lambda = 2$ is a repeated eigenvalue | A1* | Fully correct factorisation with conclusion |
| $\lambda = 8$ | B1 | |
## Part (b)
| Working | Mark | Guidance |
|---------|------|----------|
| Sets up $(A - \lambda I)\mathbf{v} = \mathbf{0}$ for $\lambda = 2$ or $\lambda = 8$ | M1 | |
| Any multiple of $\begin{pmatrix}2\\-1\\1\end{pmatrix}$ for $\lambda = 8$ | A1 | |
| Any non-zero multiple or linear combination of $\begin{pmatrix}-1\\0\\2\end{pmatrix}$ or $\begin{pmatrix}1\\2\\0\end{pmatrix}$ or $\begin{pmatrix}0\\1\\1\end{pmatrix}$ for $\lambda = 2$ | A1 | |
| Obtains a **different** linear combination or non-zero multiple of a different vector from the above options for $\lambda = 2$ | A1 | |
## Part (c)
| Working | Mark | Guidance |
|---------|------|----------|
| Forms a matrix with their eigenvectors as columns | M1 | |
| E.g. $\begin{pmatrix}-1 & 1 & 2\\ 0 & 2 & -1\\ 2 & 0 & 1\end{pmatrix}$ | A1ft | |\begin{enumerate}
\item The matrix $\mathbf { A }$ is given by
\end{enumerate}
$$\mathbf { A } = \left( \begin{array} { r r r }
6 & - 2 & 2 \\
- 2 & 3 & - 1 \\
2 & - 1 & 3
\end{array} \right)$$
(a) Show that 2 is a repeated eigenvalue of $\mathbf { A }$ and find the other eigenvalue.\\
(b) Hence find three non-parallel eigenvectors of $\mathbf { A }$.\\
(c) Find a matrix $\mathbf { P }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P }$ is a diagonal matrix.
\hfill \mbox{\textit{Edexcel FP2 2019 Q2 [11]}}