| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 11 |
| 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 techniques: finding a 3×3 matrix inverse using cofactors/adjugate method, then applying it to solve a linear system. While mechanically longer than A-level Core questions, it involves routine procedures without novel insight or tricky algebraic manipulation. The parameter 'a' adds minimal complexity since it's carried through symbolically in a standard way. |
| Spec | 4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(\det A = \Delta = 6a - 6\) | M1 | Show correct expansion process for 3 × 3 or multiply adjoint by A |
| M1 | Correct evaluation of any 2 × 2 at any stage | |
| A1 | Obtain correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(A^{-1} = \frac{1}{\Delta} \begin{pmatrix} 3a-1 & a+1 & -4 \\ 1 & 2a-1 & -2 \\ -3 & -3 & 6 \end{pmatrix}\) | M1 | Show correct process for adjoint entries |
| A1 | Obtain at least 4 correct entries in adjoint | |
| B1 | Divide by their determinant | |
| A1, 7 | Obtain completely correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{\Delta} \begin{pmatrix} 5a-7 \\ 4a-5 \\ 3 \end{pmatrix}\) | M1 | Attempt product of form \(A^{-1}C\) or eliminate to get 2 equations and solve |
| A1A1A1 ft all 3 | Obtain correct answer | |
| 4 | S.C. if det not omitted, allow max A2 ft |
**Part (i)**
$\det A = \Delta = 6a - 6$ | M1 | Show correct expansion process for 3 × 3 or multiply adjoint by **A**
| M1 | Correct evaluation of any 2 × 2 at any stage
| A1 | Obtain correct answer
---
$A^{-1} = \frac{1}{\Delta} \begin{pmatrix} 3a-1 & a+1 & -4 \\ 1 & 2a-1 & -2 \\ -3 & -3 & 6 \end{pmatrix}$ | M1 | Show correct process for adjoint entries
| A1 | Obtain at least 4 correct entries in adjoint
| B1 | Divide by their determinant
| A1, 7 | Obtain completely correct answer
---
**Part (ii)**
$\frac{1}{\Delta} \begin{pmatrix} 5a-7 \\ 4a-5 \\ 3 \end{pmatrix}$ | M1 | Attempt product of form $A^{-1}C$ or eliminate to get 2 equations and solve
| A1A1A1 ft all 3 | Obtain correct answer
| 4 | S.C. if det not omitted, allow max A2 ft
---9 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 0 & 3 & 1 \\ 1 & 1 & a \end{array} \right)$, where $a \neq 1$.\\
(i) Find $\mathbf { A } ^ { - 1 }$.\\
(ii) Hence, or otherwise, solve the equations
$$\begin{array} { r }
2 x - y + z = 1 \\
3 y + z = 2 \\
x + y + a z = 2
\end{array}$$
\hfill \mbox{\textit{OCR FP1 2010 Q9 [11]}}