Question 6 - A-Level Maths

CAIE FP1 2015 November — Question 6 10 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2015
Session	November
Marks	10
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	3x3 Matrices
Type	Find P and D for diagonalization / matrix powers
Difficulty	Standard +0.8 This is a multi-part Further Maths question requiring eigenvalue/eigenvector computation for a 3×3 matrix, verification of a given eigenvector, and matrix diagonalization including finding P^{-1}. While systematic, it involves substantial computation (3×3 matrix operations, inverse calculation) and tests understanding of diagonalization theory beyond standard A-level, placing it moderately above average difficulty.
Spec	4.03n Inverse 2x2 matrix 4.03o Inverse 3x3 matrix

6 The matrix $\mathbf { A }$, 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

## 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** | | |

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6 The matrix $\mathbf { A }$, 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 [10]}}

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