| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Prove eigenvalue/eigenvector properties |
| Difficulty | Standard +0.3 This is a straightforward Further Maths eigenvalue question with routine calculations. Part (i) is a simple algebraic proof using the definition of eigenvectors. Part (ii) requires direct computation Ae=λe for given eigenvectors. Part (iii) applies the result from (i) to construct a diagonalization, which is a standard technique. While it's Further Maths content, the question is highly structured with all eigenvectors provided, requiring no problem-solving or novel insight—just methodical application of definitions and formulas. |
| Spec | 4.03a Matrix language: terminology and notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{A}\mathbf{e} = \lambda\mathbf{e}\) and \(\mathbf{B}\mathbf{e} = \mu\mathbf{e} \Rightarrow \mathbf{A}\mathbf{e} + \mathbf{B}\mathbf{e} = \lambda\mathbf{e} + \mu\mathbf{e}\) | M1 | Adds equations |
| \(\Rightarrow (\mathbf{A} + \mathbf{B})\mathbf{e} = (\lambda + \mu)\mathbf{e}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix} 2 & 0 & 1 \\ -1 & 2 & 3 \\ 1 & 0 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 6 \\ 3 \end{pmatrix}\) | M1 | Multiplies matrix with vector |
| \(\Rightarrow \lambda = 3\) | A1 | Finds one eigenvalue |
| \(\lambda = 1, 2\) | A1 | Finds other two |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{P} = \begin{pmatrix} 1 & 1 & 0 \\ 2 & 4 & 1 \\ 1 & -1 & 0 \end{pmatrix}\) | B1 | |
| \(\mathbf{D} = \begin{pmatrix} 7^3 & 0 & 0 \\ 0 & 6^3 & 0 \\ 0 & 0 & 3^3 \end{pmatrix} = \begin{pmatrix} 343 & 0 & 0 \\ 0 & 216 & 0 \\ 0 & 0 & 27 \end{pmatrix}\) | M1 A1 | Or correctly matched permutations of columns |
## Question 5:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{A}\mathbf{e} = \lambda\mathbf{e}$ and $\mathbf{B}\mathbf{e} = \mu\mathbf{e} \Rightarrow \mathbf{A}\mathbf{e} + \mathbf{B}\mathbf{e} = \lambda\mathbf{e} + \mu\mathbf{e}$ | M1 | Adds equations |
| $\Rightarrow (\mathbf{A} + \mathbf{B})\mathbf{e} = (\lambda + \mu)\mathbf{e}$ | A1 | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix} 2 & 0 & 1 \\ -1 & 2 & 3 \\ 1 & 0 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 6 \\ 3 \end{pmatrix}$ | M1 | Multiplies matrix with vector |
| $\Rightarrow \lambda = 3$ | A1 | Finds one eigenvalue |
| $\lambda = 1, 2$ | A1 | Finds other two |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{P} = \begin{pmatrix} 1 & 1 & 0 \\ 2 & 4 & 1 \\ 1 & -1 & 0 \end{pmatrix}$ | B1 | |
| $\mathbf{D} = \begin{pmatrix} 7^3 & 0 & 0 \\ 0 & 6^3 & 0 \\ 0 & 0 & 3^3 \end{pmatrix} = \begin{pmatrix} 343 & 0 & 0 \\ 0 & 216 & 0 \\ 0 & 0 & 27 \end{pmatrix}$ | M1 A1 | Or correctly matched permutations of columns |
---5 It is given that $\lambda$ is an eigenvalue of the matrix $\mathbf { A }$ with $\mathbf { e }$ as a corresponding eigenvector, and $\mu$ is an eigenvalue of the matrix $\mathbf { B }$ for which $\mathbf { e }$ is also a corresponding eigenvector.\\
(i) Show that $\lambda + \mu$ is an eigenvalue of the matrix $\mathbf { A } + \mathbf { B }$ with $\mathbf { e }$ as a corresponding eigenvector.\\
The matrix $\mathbf { A }$, given by
$$\mathbf { A } = \left( \begin{array} { r r r }
2 & 0 & 1 \\
- 1 & 2 & 3 \\
1 & 0 & 2
\end{array} \right)$$
has $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ as eigenvectors.\\
(ii) Find the corresponding eigenvalues.\\
The matrix $\mathbf { B }$ has eigenvalues 4, 5 and 1 with corresponding eigenvectors $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ respectively.\\
(iii) Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }$.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q5}}