| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Prove eigenvalue/eigenvector properties |
| Difficulty | Standard +0.8 This is a two-part Further Maths question requiring proof of eigenvalue properties followed by diagonalization. The proof part (i-ii) requires understanding of eigenvalue definitions and matrix invertibility but follows standard logical steps. The computational part requires finding eigenvalues of an upper triangular matrix (straightforward), then applying the proven result to find eigenvalues of B, finding eigenvectors, and constructing the diagonalization. While multi-step, each component uses standard Further Maths techniques without requiring novel insight. |
| Spec | 4.03a Matrix language: terminology and notation4.03l Singular/non-singular matrices4.03o Inverse 3x3 matrix4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1) |
The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that if $\mathbf { A }$ is non-singular then\\
(i) $\lambda \neq 0$,\\
(ii) the matrix $\mathbf { A } ^ { - 1 }$ has $\lambda ^ { - 1 }$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector.
The $3 \times 3$ matrices $\mathbf { A }$ and $\mathbf { B }$ are given by
$$\mathbf { A } = \left( \begin{array} { r r r }
- 2 & 2 & - 4 \\
0 & - 1 & 5 \\
0 & 0 & 3
\end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 3 \mathbf { I } ) ^ { - 1 }$$
where $\mathbf { I }$ is the $3 \times 3$ identity matrix. Find a non-singular matrix $\mathbf { P }$, and a diagonal matrix $\mathbf { D }$, such that $\mathbf { B } = \mathbf { P D P } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE FP1 2014 Q11 OR}}