| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Find inverse then solve system |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring standard 3×3 matrix inversion (using cofactors/adjugate method) and solving a system of linear equations using the inverse. While the calculations are lengthy and require care, the techniques are routine for FP3 students with no novel problem-solving required. The 14 marks reflect computational work rather than conceptual difficulty. |
| Spec | 4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(A(x) = \begin{pmatrix} 1 & x & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}\) | ||
| Cofactors: \(\begin{pmatrix} -2 & 2 & 3 \\ -1 & 1 & x-1 \\ 2x & -5 & -3x \end{pmatrix}\) | M1A1, A1, A1 | |
| Determinant \(= 2x - 3 - 2 = 2x - 5\) | M1A1 | |
| \(A^{-1}(x) = \frac{1}{2x-5}\begin{pmatrix} -2 & -1 & 2x \\ 2 & 1 & -5 \\ 3 & x-1 & -3x \end{pmatrix}\) | M1A1ft | (8) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix} p \\ q \\ r \end{pmatrix} = B^{-1}\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} = \frac{1}{1}\begin{pmatrix} -2 & -1 & 6 \\ 2 & 1 & -5 \\ 3 & 2 & -9 \end{pmatrix}\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}\) | M1A1ft | |
| \(= (17, -13, -24)\) | M1A1 | (4) |
## (a)
$A(x) = \begin{pmatrix} 1 & x & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}$ | |
Cofactors: $\begin{pmatrix} -2 & 2 & 3 \\ -1 & 1 & x-1 \\ 2x & -5 & -3x \end{pmatrix}$ | M1A1, A1, A1 |
Determinant $= 2x - 3 - 2 = 2x - 5$ | M1A1 |
$A^{-1}(x) = \frac{1}{2x-5}\begin{pmatrix} -2 & -1 & 2x \\ 2 & 1 & -5 \\ 3 & x-1 & -3x \end{pmatrix}$ | M1A1ft | (8)
## (b)
$\begin{pmatrix} p \\ q \\ r \end{pmatrix} = B^{-1}\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} = \frac{1}{1}\begin{pmatrix} -2 & -1 & 6 \\ 2 & 1 & -5 \\ 3 & 2 & -9 \end{pmatrix}\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$ | M1A1ft |
$= (17, -13, -24)$ | M1A1 | (4)
---$$\mathbf{A}(x) = \begin{pmatrix} 1 & x & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}, \quad x \neq \frac{5}{2}$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the inverse of $\mathbf{A}(x)$.
$$\mathbf{B} = \begin{pmatrix} 1 & 3 & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}$$ [8]
The image of the vector $\begin{pmatrix} p \\ q \\ r \end{pmatrix}$ when transformed by $\mathbf{B}$ is $\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$
\item Find the values of $p$, $q$ and $r$. [4]
\end{enumerate}
(Total 14 marks)
\hfill \mbox{\textit{Edexcel FP3 Q7 [12]}}