Question 10 - A-Level Maths

CAIE FP1 2003 November — Question 10 12 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2003
Session	November
Marks	12
Paper	Download PDF ↗
Topic	Invariant lines and eigenvalues and vectors
Type	Find P and D for A² = PDP⁻¹ or A⁻¹ = PDP⁻¹
Difficulty	Standard +0.8 This is a substantial Further Maths question requiring eigenvalue/eigenvector calculation for a 3×3 matrix (solving a cubic characteristic equation), then applying diagonalization theory to express a matrix polynomial. The 'hence' part requires recognizing that A + A² + A³ = P(D + D² + D³)P⁻¹, which demands conceptual understanding beyond routine computation. Worth 12 marks total with multiple computational steps and theoretical insight.
Spec	4.03n Inverse 2x2 matrix 4.03o Inverse 3x3 matrix

Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf{A}$, where $$\mathbf{A} = \begin{pmatrix} 6 & 4 & 1 \\ -6 & -1 & 3 \\ 8 & 8 & 4 \end{pmatrix}.$$ [8] Hence find a non-singular matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{A} + \mathbf{A}^2 + \mathbf{A}^3 = \mathbf{PDP}^{-1}$. [4]

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Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf{A}$, where
$$\mathbf{A} = \begin{pmatrix} 6 & 4 & 1 \\ -6 & -1 & 3 \\ 8 & 8 & 4 \end{pmatrix}.$$ [8]

Hence find a non-singular matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{A} + \mathbf{A}^2 + \mathbf{A}^3 = \mathbf{PDP}^{-1}$. [4]

\hfill \mbox{\textit{CAIE FP1 2003 Q10 [12]}}

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