| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Eigenvalues and eigenvectors |
| Difficulty | Standard +0.3 This is a standard Further Pure 3 matrices question covering routine techniques: determinant calculation (cofactor expansion), matrix inversion (adjugate method), and finding eigenvalues/eigenvectors. Part (a) is straightforward computation, part (b) is mechanical but lengthy (6 marks reflects workload not difficulty), part (c) is trivial matrix multiplication, and part (d) requires solving (A - 8I)v = 0 which is standard row reduction. All techniques are textbook exercises with no novel insight required, making this slightly easier than average even for FP3 standard. |
| Spec | 4.03j Determinant 3x3: calculation4.03o Inverse 3x3 matrix |
$$\mathbf{A} = \begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & k \end{pmatrix}.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\det \mathbf{A} = 20 - 4k$. [2]
\item Find $\mathbf{A}^{-1}$. [6]
\end{enumerate}
Given that $k = 3$ and that $\begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}$ is an eigenvector of $\mathbf{A}$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the corresponding eigenvalue. [2]
\end{enumerate}
Given that the only other distinct eigenvalue of $\mathbf{A}$ is $8$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find a corresponding eigenvector. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q37 [14]}}