| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| 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 +1.8 This Further Maths question combines eigenvalue theory with matrix diagonalization and series convergence. Parts (i)-(ii) test conceptual understanding of eigenvectors (trivial), part (iii) requires finding P and D for a triangular matrix (straightforward since eigenvalues are on diagonal), but part (iv) involves manipulating an infinite series of matrix powers and applying convergence conditions, requiring sophisticated algebraic manipulation and understanding of when geometric series converge—this elevates it significantly above routine exercises. |
| Spec | 4.03a Matrix language: terminology and notation4.03o Inverse 3x3 matrix4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1) |
It is given that $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A }$, with corresponding eigenvalue $\lambda$.\\
(i) Write down another eigenvector of $\mathbf { A }$ corresponding to $\lambda$.\\
(ii) Write down an eigenvector and corresponding eigenvalue of $\mathbf { A } ^ { n }$, where $n$ is a positive integer.\\
Let $\mathbf { A } = \left( \begin{array} { l l l } 3 & 0 & 0 \\ 2 & 7 & 0 \\ 4 & 8 & 1 \end{array} \right)$.\\
(iii) Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.\\
(iv) Determine the set of values of the real constant $k$ such that
$$\sum _ { n = 1 } ^ { \infty } k ^ { n } \left( \mathbf { A } ^ { n } - k \mathbf { A } ^ { n + 1 } \right) = k \mathbf { A } .$$
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q11 OR}}