| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Find P and D for diagonalization / matrix powers |
| Difficulty | Standard +0.3 This is a structured FP3 eigenvector/eigenvalue question with significant scaffolding. Part (a) is routine verification (matrix multiplication), part (b) requires solving (A - 9I)v = 0 which is standard, and part (c) involves assembling P and D from given information. While 3×3 matrices are Further Maths content, the question guides students through each step without requiring novel insight or difficult algebraic manipulation. |
| Spec | 4.03a Matrix language: terminology and notation |
$$\mathbf{A} = \begin{pmatrix} 1 & 0 & 4 \\ 0 & 5 & 4 \\ 4 & 4 & 3 \end{pmatrix}.$$
\begin{enumerate}[label=(\alph*)]
\item Verify that $\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$ is an eigenvector of $\mathbf{A}$ and find the corresponding eigenvalue. [3]
\item Show that $9$ is another eigenvalue of $\mathbf{A}$ and find the corresponding eigenvector. [5]
\item Given that the third eigenvector of $\mathbf{A}$ is $\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$, write down a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that
$$\mathbf{P}^T\mathbf{A}\mathbf{P} = \mathbf{D}.$$ [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q8 [13]}}