Question 10 - A-Level Maths

CAIE FP1 2015 June — Question 10 12 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2015
Session	June
Marks	12
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Invariant lines and eigenvalues and vectors
Type	Find P and D for A = PDP⁻¹
Difficulty	Challenging +1.2 This is a structured multi-part eigenvalue question with significant scaffolding. Students are given one eigenvector to find its eigenvalue (routine multiplication), told the other two eigenvalues to find eigenvectors (solving linear systems), then construct the diagonalization. The final part about matrix B requires understanding that similar matrices share eigenvalues but transform eigenvectors, which is conceptually slightly above routine. Overall, this is a standard Further Maths question testing core diagonalization techniques with more guidance than problem-solving required.
Spec	4.03a Matrix language: terminology and notation 4.03h Determinant 2x2: calculation 4.03i Determinant: area scale factor and orientation 4.03n Inverse 2x2 matrix 4.03o Inverse 3x3 matrix

10 The matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 2 & - 3 \\ 2 & 2 & 3 \\ - 3 & 3 & 3 \end{array} \right)$$ The matrix $\mathbf { A }$ has an eigenvector $\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)$. Find the corresponding eigenvalue. The matrix $\mathbf { A }$ also has eigenvalues 4 and 6. Find corresponding eigenvectors. Hence find a matrix $\mathbf { P }$ such that $\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$, where $\mathbf { D }$ is a diagonal matrix which is to be determined. The matrix $\mathbf { B }$ is such that $\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }$, where $$\mathbf { Q } = \left( \begin{array} { r r r } 4 & 11 & 5 \\ 1 & 4 & 2 \\ 1 & 2 & 1 \end{array} \right)$$ By using the expression $\mathbf { P D P } ^ { - 1 }$ for $\mathbf { A }$, find the set of eigenvalues and a corresponding set of eigenvectors for $\mathbf { B }$.
[0pt] [Question 11 is printed on the next page.]

Show mark scheme Show mark scheme source

Question 10:

Eigenvalue verification:

Answer	Marks	Guidance
$\begin{pmatrix} 2 & 2 & -3 \\ 2 & 2 & 3 \\ -3 & 3 & 3 \end{pmatrix}\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix} = \begin{pmatrix} -3 \\ 3 \\ -3 \end{pmatrix} \Rightarrow$ eigenvalue is $-3$	M1A1 (2)	—

Eigenvector for $\lambda = 4$:

Answer	Marks	Guidance
$\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -2 & 2 & -3 \\ -3 & 3 & -1 \end{vmatrix} = \begin{pmatrix} 7 \\ 7 \\ 0 \end{pmatrix} \sim \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ (Or by equations)	M1A1	—

Eigenvector for $\lambda = 6$:

Answer	Marks	Guidance
$\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -4 & 2 & -3 \\ -3 & 3 & -3 \end{vmatrix} = \begin{pmatrix} 3 \\ -3 \\ -6 \end{pmatrix} \sim \begin{pmatrix} 1 \\ -1 \\ -2 \end{pmatrix}$	A1 (3)	—

Modal and diagonal matrices:

Answer	Marks	Guidance
$\mathbf{P} = \begin{pmatrix} 1 & 1 & 1 \\ -1 & 1 & -1 \\ 1 & 0 & -2 \end{pmatrix}$, $\mathbf{D} = \begin{pmatrix} -3 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 6 \end{pmatrix}$	B1$\sqrt{}$B1$\sqrt{}$ (2), M1;A1	—

Expression for B:

Answer	Marks	Guidance
$\mathbf{B} = \mathbf{QAQ}^{-1} = \mathbf{QPAP}^{-1}\mathbf{Q}^{-1} = \mathbf{QPA}(\mathbf{QP})^{-1}$	B1	—

QP calculation:

Answer	Marks	Guidance
$\mathbf{QP} = \begin{pmatrix} -2 & 15 & -17 \\ -1 & 5 & -7 \\ 0 & 3 & -3 \end{pmatrix}$ (CAO)	—	—

Eigenvalues of B:

Answer	Marks	Guidance
Eigenvalues are $-3$, $4$ and $6$ (same as those for $\mathbf{A}$). (CAO)	B1	—

Eigenvectors of B:

Answer	Marks	Guidance
$\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 15 \\ 5 \\ 3 \end{pmatrix}, \begin{pmatrix} 17 \\ 7 \\ 3 \end{pmatrix}$	B1 (5), Total: 12	Columns of QP

# Question 10:

**Eigenvalue verification:**
$\begin{pmatrix} 2 & 2 & -3 \\ 2 & 2 & 3 \\ -3 & 3 & 3 \end{pmatrix}\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix} = \begin{pmatrix} -3 \\ 3 \\ -3 \end{pmatrix} \Rightarrow$ eigenvalue is $-3$ | M1A1 (2) | —

**Eigenvector for $\lambda = 4$:**
$\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -2 & 2 & -3 \\ -3 & 3 & -1 \end{vmatrix} = \begin{pmatrix} 7 \\ 7 \\ 0 \end{pmatrix} \sim \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ (Or by equations) | M1A1 | —

**Eigenvector for $\lambda = 6$:**
$\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -4 & 2 & -3 \\ -3 & 3 & -3 \end{vmatrix} = \begin{pmatrix} 3 \\ -3 \\ -6 \end{pmatrix} \sim \begin{pmatrix} 1 \\ -1 \\ -2 \end{pmatrix}$ | A1 (3) | —

**Modal and diagonal matrices:**
$\mathbf{P} = \begin{pmatrix} 1 & 1 & 1 \\ -1 & 1 & -1 \\ 1 & 0 & -2 \end{pmatrix}$, $\mathbf{D} = \begin{pmatrix} -3 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 6 \end{pmatrix}$ | B1$\sqrt{}$B1$\sqrt{}$ (2), M1;A1 | —

**Expression for B:**
$\mathbf{B} = \mathbf{QAQ}^{-1} = \mathbf{QPAP}^{-1}\mathbf{Q}^{-1} = \mathbf{QPA}(\mathbf{QP})^{-1}$ | B1 | —

**QP calculation:**
$\mathbf{QP} = \begin{pmatrix} -2 & 15 & -17 \\ -1 & 5 & -7 \\ 0 & 3 & -3 \end{pmatrix}$ (CAO) | — | —

**Eigenvalues of B:**
Eigenvalues are $-3$, $4$ and $6$ (same as those for $\mathbf{A}$). (CAO) | B1 | —

**Eigenvectors of B:**
$\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 15 \\ 5 \\ 3 \end{pmatrix}, \begin{pmatrix} 17 \\ 7 \\ 3 \end{pmatrix}$ | B1 (5), **Total: 12** | Columns of QP

---

Show LaTeX source

10 The matrix $\mathbf { A }$ is given by

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

The matrix $\mathbf { A }$ has an eigenvector $\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)$. Find the corresponding eigenvalue.

The matrix $\mathbf { A }$ also has eigenvalues 4 and 6. Find corresponding eigenvectors.

Hence find a matrix $\mathbf { P }$ such that $\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$, where $\mathbf { D }$ is a diagonal matrix which is to be determined.

The matrix $\mathbf { B }$ is such that $\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }$, where

$$\mathbf { Q } = \left( \begin{array} { r r r } 
4 & 11 & 5 \\
1 & 4 & 2 \\
1 & 2 & 1
\end{array} \right)$$

By using the expression $\mathbf { P D P } ^ { - 1 }$ for $\mathbf { A }$, find the set of eigenvalues and a corresponding set of eigenvectors for $\mathbf { B }$.\\[0pt]
[Question 11 is printed on the next page.]

\hfill \mbox{\textit{CAIE FP1 2015 Q10 [12]}}

This paper (12 questions)

View full paper

Q1 4 Q2 6 Q3 7 Q4 8 Q5 9 Q6 9 Q7 9 Q8 11 Q9 11 Q10 12 Q11 EITHER Q11 OR