| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Orthogonal matrix diagonalization |
| Difficulty | Standard +0.3 This is a standard FP3 eigenvalue/eigenvector question with routine diagonalization. The matrix is symmetric with simple integer eigenvalues (2 and 4), eigenvectors are straightforward to find and normalize, and the geometric interpretation (stretch along perpendicular axes) follows directly from the diagonal form. While it's Further Maths content, it requires only systematic application of learned procedures without novel insight. |
| Spec | 4.03h Determinant 2x2: calculation4.03l Singular/non-singular matrices |
The transformation $R$ is represented by the matrix $\mathbf{A}$, where
$$\mathbf{A} = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}.$$
\begin{enumerate}[label=(\alph*)]
\item Find the eigenvectors of $\mathbf{A}$. [5]
\item Find an orthogonal matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that
$$\mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}.$$ [5]
\item Hence describe the transformation $R$ as a combination of geometrical transformations, stating clearly their order. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q28 [14]}}