| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Eigenvalues and eigenvectors |
| Difficulty | Standard +0.3 This is a structured multi-part question on eigenvalues/eigenvectors that guides students through standard procedures. Part (a) uses the given eigenvalue to find k by substituting into det(M - 3I) = 0, part (b) finds remaining eigenvalues from the characteristic equation, and part (c) solves a standard system for the eigenvector. While it involves 3×3 matrices (Further Maths content), the question is methodical with clear signposting and no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts determinant of \(\mathbf{M} - 3\mathbf{I}\) | M1 | Should be a recognisable attempt at the determinant (if in doubt, at least 2 "terms" should be correct) |
| \(0 - k + 2(-k+3) = 0 \Rightarrow k = 2\) | dM1 | Puts \(= 0\) (may be implied) and solves for \(k\). Dependent on first M |
| \(k = 2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\det(\mathbf{M} - \lambda\mathbf{I}) = (3-\lambda)\left[-\lambda(1-\lambda)-k\right] - k\left[-1(1-\lambda)-1\right] + 2(-k+\lambda)\) | M1 | Attempts determinant of \(\mathbf{M} - \lambda\mathbf{I}\) (may be seen in (a)) |
| Uses \(k=2\), sets \(= 0\) | dM1 | Uses their \(k\) in determinant and puts \(= 0\). Dependent on first M |
| \(\{(3-\lambda)\}(\lambda^2 - \lambda - 2) = 0 \Rightarrow \lambda = \ldots\) | ddM1 | Solves to find the 2 other eigenvalues. Dependent on both previous M's |
| \(\lambda = -1, 2\) | A1 | Correct eigenvalues. Must follow \(k=2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Forms equations using eigenvalue 3, e.g. \(3x + ``2"y + 2z = 3x\), \(-x+z=3y\), \(x+``2"y+z=3z\) | M1 | Expands to obtain at least 2 equations. Allow if \(k\) is present |
| \(k\begin{pmatrix}4\\-1\\1\end{pmatrix}\) or \(k(4\mathbf{i} - \mathbf{j} + \mathbf{k})\) | A1 | Any non-zero multiple but must be a vector |
# Question 3(a) [Eigenvalue/Matrix]:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts determinant of $\mathbf{M} - 3\mathbf{I}$ | M1 | Should be a recognisable attempt at the determinant (if in doubt, at least 2 "terms" should be correct) |
| $0 - k + 2(-k+3) = 0 \Rightarrow k = 2$ | dM1 | Puts $= 0$ (may be implied) and solves for $k$. **Dependent on first M** |
| $k = 2$ | A1 | |
---
# Question 3(b) [Characteristic Equation]:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\det(\mathbf{M} - \lambda\mathbf{I}) = (3-\lambda)\left[-\lambda(1-\lambda)-k\right] - k\left[-1(1-\lambda)-1\right] + 2(-k+\lambda)$ | M1 | Attempts determinant of $\mathbf{M} - \lambda\mathbf{I}$ (may be seen in (a)) |
| Uses $k=2$, sets $= 0$ | dM1 | Uses their $k$ in determinant and puts $= 0$. **Dependent on first M** |
| $\{(3-\lambda)\}(\lambda^2 - \lambda - 2) = 0 \Rightarrow \lambda = \ldots$ | ddM1 | Solves to find the 2 other eigenvalues. **Dependent on both previous M's** |
| $\lambda = -1, 2$ | A1 | Correct eigenvalues. Must follow $k=2$ |
---
# Question 3(c) [Eigenvector]:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Forms equations using eigenvalue 3, e.g. $3x + ``2"y + 2z = 3x$, $-x+z=3y$, $x+``2"y+z=3z$ | M1 | Expands to obtain at least 2 equations. Allow if $k$ is present |
| $k\begin{pmatrix}4\\-1\\1\end{pmatrix}$ or $k(4\mathbf{i} - \mathbf{j} + \mathbf{k})$ | A1 | Any non-zero multiple but must be a vector |
---3. $\mathbf { M } = \left( \begin{array} { r r r } 3 & k & 2 \\ - 1 & 0 & 1 \\ 1 & k & 1 \end{array} \right)$, where $k$ is a constant
Given that 3 is an eigenvalue of $\mathbf { M }$,
\begin{enumerate}[label=(\alph*)]
\item find the value of $k$.
\item Hence find the other two eigenvalues of $\mathbf { M }$.
\item Find an eigenvector corresponding to the eigenvalue 3\\
3. $\quad \mathbf { M } = \left( \begin{array} { r c c } 3 & k & 2 \\ - 1 & 0 & 1 \\ 1 & k & 1 \end{array} \right)$, where $k$ is a constant Given that 3 is an eigenvalue of $\mathbf { M }$, (a) find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2018 Q3 [9]}}