CAIE FP1 2012 November — Question 10 13 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionNovember
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicInvariant lines and eigenvalues and vectors
TypeFind P and D for A = PDP⁻¹
DifficultyStandard +0.8 This is a multi-part Further Maths question requiring eigenvalue/eigenvector calculation for a 3×3 upper triangular matrix (straightforward), constructing diagonalization P and D, finding P⁻¹ (3×3 matrix inversion), computing A^n, and evaluating a limit involving matrix powers. While upper triangular makes eigenvalues trivial to read off, the 3×3 eigenvector calculations, matrix inversion, and final limit require sustained careful work across multiple techniques, placing it moderately above average difficulty.
Spec4.03i Determinant: area scale factor and orientation4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix
10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16 \\ 0 & 2 & 3 \\ 0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let \(n\) be a positive integer. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$ Find \(\mathbf { P } ^ { - 1 }\) and \(\mathbf { A } ^ { n }\). Hence find \(\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)\).
Question 10:
Eigenvalues (1 mark):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Eigenvalues are \(1, 2, 3\)B1 1
Eigenvectors (4 marks):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\lambda=1\): \(\mathbf{e}_1=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\0&4&-16\\0&1&3\end{vmatrix}=\begin{pmatrix}28\\0\\0\end{pmatrix}\sim\begin{pmatrix}1\\0\\0\end{pmatrix}\)M1A1 Finds eigenvectors
\(\lambda=2\): \(\mathbf{e}_2=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\-1&4&-16\\0&0&1\end{vmatrix}=\begin{pmatrix}4\\1\\0\end{pmatrix}\)A1
\(\lambda=3\): \(\mathbf{e}_3=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\-2&4&-16\\0&-1&3\end{vmatrix}=\begin{pmatrix}-4\\6\\2\end{pmatrix}\sim\begin{pmatrix}-2\\3\\1\end{pmatrix}\)A1 4
P and D matrices (2 marks):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\mathbf{P}=\begin{pmatrix}1&4&-2\\0&1&3\\0&0&1\end{pmatrix}\)B1 States P
\(\mathbf{D}=\begin{pmatrix}1&0&0\\0&2^n&0\\0&0&3^n\end{pmatrix}\)B1 2
Finding \(\mathbf{A}^n\) (5 marks):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Det \(\mathbf{P}=1 \Rightarrow\) Adj \(\mathbf{P}=\mathbf{P}^{-1}=\begin{pmatrix}1&-4&14\\0&1&-3\\0&0&1\end{pmatrix}\)M1A1 Finds inverse of P
\(\mathbf{A}^n=\mathbf{PDP}^{-1}\)
\(=\mathbf{P}\begin{pmatrix}1&-4&14\\0&2^n&-3\cdot2^n\\0&0&3^n\end{pmatrix}\)M1A1 Finds \(\mathbf{A}^n\)
\(=\begin{pmatrix}1&[-4+4\cdot2^n]&[14-12\cdot2^n-2\cdot3^n]\\0&2^n&[-3\cdot2^n+3^{n+1}]\\0&0&3^n\end{pmatrix}\)A1 5
Limit (1 mark):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(3^{-n}\mathbf{A}^n \to \begin{pmatrix}0&0&-2\\0&0&3\\0&0&1\end{pmatrix}\) as \(n\to\infty\)B1 1 
# Question 10:

## Eigenvalues (1 mark):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Eigenvalues are $1, 2, 3$ | B1 | 1 | States eigenvalues |

## Eigenvectors (4 marks):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\lambda=1$: $\mathbf{e}_1=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\0&4&-16\\0&1&3\end{vmatrix}=\begin{pmatrix}28\\0\\0\end{pmatrix}\sim\begin{pmatrix}1\\0\\0\end{pmatrix}$ | M1A1 | Finds eigenvectors |
| $\lambda=2$: $\mathbf{e}_2=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\-1&4&-16\\0&0&1\end{vmatrix}=\begin{pmatrix}4\\1\\0\end{pmatrix}$ | A1 | |
| $\lambda=3$: $\mathbf{e}_3=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\-2&4&-16\\0&-1&3\end{vmatrix}=\begin{pmatrix}-4\\6\\2\end{pmatrix}\sim\begin{pmatrix}-2\\3\\1\end{pmatrix}$ | A1 | 4 | |

## P and D matrices (2 marks):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{P}=\begin{pmatrix}1&4&-2\\0&1&3\\0&0&1\end{pmatrix}$ | B1 | States **P** |
| $\mathbf{D}=\begin{pmatrix}1&0&0\\0&2^n&0\\0&0&3^n\end{pmatrix}$ | B1 | 2 | States **D** |

## Finding $\mathbf{A}^n$ (5 marks):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Det $\mathbf{P}=1 \Rightarrow$ Adj $\mathbf{P}=\mathbf{P}^{-1}=\begin{pmatrix}1&-4&14\\0&1&-3\\0&0&1\end{pmatrix}$ | M1A1 | Finds inverse of **P** |
| $\mathbf{A}^n=\mathbf{PDP}^{-1}$ | | |
| $=\mathbf{P}\begin{pmatrix}1&-4&14\\0&2^n&-3\cdot2^n\\0&0&3^n\end{pmatrix}$ | M1A1 | Finds $\mathbf{A}^n$ |
| $=\begin{pmatrix}1&[-4+4\cdot2^n]&[14-12\cdot2^n-2\cdot3^n]\\0&2^n&[-3\cdot2^n+3^{n+1}]\\0&0&3^n\end{pmatrix}$ | A1 | 5 | |

## Limit (1 mark):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $3^{-n}\mathbf{A}^n \to \begin{pmatrix}0&0&-2\\0&0&3\\0&0&1\end{pmatrix}$ as $n\to\infty$ | B1 | 1 | **[13]** |

---
10 Write down the eigenvalues of the matrix $\mathbf { A }$, where

$$\mathbf { A } = \left( \begin{array} { r r r } 
1 & 4 & - 16 \\
0 & 2 & 3 \\
0 & 0 & 3
\end{array} \right)$$

Find corresponding eigenvectors.

Let $n$ be a positive integer. Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that

$$\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$

Find $\mathbf { P } ^ { - 1 }$ and $\mathbf { A } ^ { n }$.

Hence find $\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)$.

\hfill \mbox{\textit{CAIE FP1 2012 Q10 [13]}}
This paper (12 questions)
View full paper
Q1 5 Q2 6 Q3 6 Q4 8 Q5 8 Q6 9 Q7 9 Q8 10 Q9 12 Q10 13 Q11 EITHER Q11 OR