| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find constant from eigenvalue condition |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring: (1) finding a constant using the characteristic equation with a known eigenvalue, (2) finding eigenvectors by solving homogeneous systems, (3) proving invariance of eigenspaces, and (4) finding a vector perpendicular to a plane via cross product and checking if it's an eigenvector. While each individual step uses standard techniques, the question requires sustained reasoning across multiple concepts (eigenvalues, eigenvectors, linear spaces, orthogonality) and the final part requires some geometric insight about the relationship between eigenspaces and perpendicular vectors. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative) |
One of the eigenvalues of the matrix
$$\mathbf { A } = \left( \begin{array} { r r r }
1 & - 4 & 6 \\
2 & - 4 & 2 \\
- 3 & 4 & a
\end{array} \right)$$
is - 2 . Find the value of $a$.
Another eigenvalue of $\mathbf { A }$ is - 5 . Find eigenvectors $\mathbf { e } _ { 1 }$ and $\mathbf { e } _ { 2 }$ corresponding to the eigenvalues - 2 and - 5 respectively.
The linear space spanned by $\mathbf { e } _ { 1 }$ and $\mathbf { e } _ { 2 }$ is denoted by $V$.\\
(i) Prove that, for any vector $\mathbf { x }$ belonging to $V$, the vector $\mathbf { A x }$ also belongs to $V$.\\
(ii) Find a non-zero vector which is perpendicular to every vector in $V$, and determine whether it is an eigenvector of $\mathbf { A }$.
\hfill \mbox{\textit{CAIE FP1 2009 Q11 OR}}