| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Paper | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Prove eigenvalue/eigenvector properties |
| Difficulty | Standard +0.8 This question combines theoretical proof (showing eigenvector properties under matrix operations) with computational work (finding eigenvalues/eigenvectors of 3×3 matrices). The proofs are straightforward applications of definitions, but finding eigenvalues requires solving a cubic characteristic equation and then finding corresponding eigenvectors through row reduction. The B² part adds complexity as students must either compute B² first or use the proven relationship. This is moderately challenging for Further Maths students but follows standard techniques. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar |
The vector $\mathbf { e }$ is an eigenvector of the square matrix $\mathbf { G }$. Show that\\
(i) $\mathbf { e }$ is an eigenvector of $\mathbf { G } + k \mathbf { I }$, where $k$ is a scalar and $\mathbf { I }$ is an identity matrix,\\
(ii) $\mathbf { e }$ is an cigenvector of $\mathbf { G } ^ { 2 }$.
Find the eigenvalues, and corresponding eigenvectors, of the matrices $\mathbf { A }$ and $\mathbf { B } ^ { 2 }$, where
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & - 3 & 0 \\
1 & 0 & 1 \\
- 1 & 3 & 2
\end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r }
- 5 & - 3 & 0 \\
1 & - 8 & 1 \\
- 1 & 3 & - 6
\end{array} \right)$$
\hfill \mbox{\textit{CAIE FP1 2002 Q11 EITHER}}