Question 6 - A-Level Maths

CAIE FP1 2015 November — Question 6

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2015
Session	November
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Invariant lines and eigenvalues and vectors
Type	Find P and D for A = PDP⁻¹
Difficulty	Standard +0.3 This is a standard Further Maths eigenvalue/eigenvector question with straightforward calculations. Finding eigenvectors from given eigenvalues, verifying a given eigenvector, and constructing the diagonalization are all routine procedures requiring no novel insight, though the 3×3 matrix adds some computational work compared to typical A-level questions.
Spec	4.03a Matrix language: terminology and notation 4.03n Inverse 2x2 matrix 4.03o Inverse 3x3 matrix

6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that $\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$. Find the corresponding eigenvalue. Find a diagonal matrix $\mathbf { D }$ and matrices $\mathbf { P }$ and $\mathbf { P } ^ { - 1 }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }$.

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Question 6:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$\lambda=1$: $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 10 & -8 & 10 \\ 7 & -5 & 7 \end{vmatrix} = \begin{pmatrix}-6\\0\\6\end{pmatrix} \sim \begin{pmatrix}1\\0\\-1\end{pmatrix}$	M1A1	oe
$\lambda=3$: $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -2 & 0 & 0 \\ 10 & -10 & 10 \end{vmatrix} = \begin{pmatrix}0\\20\\20\end{pmatrix} \sim \begin{pmatrix}0\\1\\1\end{pmatrix}$	A1	oe
$\begin{pmatrix}1 & 0 & 0\\10 & -7 & 10\\7 & -5 & 8\end{pmatrix}\begin{pmatrix}0\\2\\1\end{pmatrix} = \begin{pmatrix}0\\-4\\-2\end{pmatrix} = -2\begin{pmatrix}0\\2\\1\end{pmatrix} \Rightarrow \lambda = -2$	M1A1
$\mathbf{D} = \begin{pmatrix}-2 & 0 & 0\\0 & 1 & 0\\0 & 0 & 3\end{pmatrix}$, $\mathbf{P} = \begin{pmatrix}0 & 1 & 0\\2 & 0 & 1\\1 & -1 & 1\end{pmatrix}$	B1$\checkmark$ B1$\checkmark$	or other multiples or permutations
$\det \mathbf{P} = -1$ (or 1 depending on permutation)	B1
$\text{Adj } \mathbf{P} = \begin{pmatrix}1 & -1 & 1\\-1 & 0 & 0\\-2 & 1 & 2\end{pmatrix} \Rightarrow \mathbf{P}^{-1} = \begin{pmatrix}-1 & 1 & -1\\1 & 0 & 0\\2 & -1 & 2\end{pmatrix}$	M1A1	or other permutations
Total: 10 marks	[10]

## Question 6:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\lambda=1$: $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 10 & -8 & 10 \\ 7 & -5 & 7 \end{vmatrix} = \begin{pmatrix}-6\\0\\6\end{pmatrix} \sim \begin{pmatrix}1\\0\\-1\end{pmatrix}$ | M1A1 | oe |
| $\lambda=3$: $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -2 & 0 & 0 \\ 10 & -10 & 10 \end{vmatrix} = \begin{pmatrix}0\\20\\20\end{pmatrix} \sim \begin{pmatrix}0\\1\\1\end{pmatrix}$ | A1 | oe |
| $\begin{pmatrix}1 & 0 & 0\\10 & -7 & 10\\7 & -5 & 8\end{pmatrix}\begin{pmatrix}0\\2\\1\end{pmatrix} = \begin{pmatrix}0\\-4\\-2\end{pmatrix} = -2\begin{pmatrix}0\\2\\1\end{pmatrix} \Rightarrow \lambda = -2$ | M1A1 | |
| $\mathbf{D} = \begin{pmatrix}-2 & 0 & 0\\0 & 1 & 0\\0 & 0 & 3\end{pmatrix}$, $\mathbf{P} = \begin{pmatrix}0 & 1 & 0\\2 & 0 & 1\\1 & -1 & 1\end{pmatrix}$ | B1$\checkmark$ B1$\checkmark$ | or other multiples or permutations |
| $\det \mathbf{P} = -1$ (or 1 depending on permutation) | B1 | |
| $\text{Adj } \mathbf{P} = \begin{pmatrix}1 & -1 & 1\\-1 & 0 & 0\\-2 & 1 & 2\end{pmatrix} \Rightarrow \mathbf{P}^{-1} = \begin{pmatrix}-1 & 1 & -1\\1 & 0 & 0\\2 & -1 & 2\end{pmatrix}$ | M1A1 | or other permutations |
| **Total: 10 marks** | [10] | |

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6 The matrix A, where

$$\mathbf { A } = \left( \begin{array} { r r r } 
1 & 0 & 0 \\
10 & - 7 & 10 \\
7 & - 5 & 8
\end{array} \right)$$

has eigenvalues 1 and 3. Find corresponding eigenvectors.

It is given that $\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$. Find the corresponding eigenvalue.

Find a diagonal matrix $\mathbf { D }$ and matrices $\mathbf { P }$ and $\mathbf { P } ^ { - 1 }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }$.

\hfill \mbox{\textit{CAIE FP1 2015 Q6}}

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Q1 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 EITHER Q11 OR