OCR FP1 2011 June — Question 6 7 marks

Exam BoardOCR
ModuleFP1 (Further Pure Mathematics 1)
Year2011
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
Topic3x3 Matrices
TypeMatrix inverse calculation
DifficultyStandard +0.3 This is a straightforward 3×3 matrix inversion problem from Further Pure 1, requiring systematic application of the cofactor method or row reduction. While it involves more computation than a 2×2 inverse, it's a standard textbook exercise with no conceptual challenges beyond careful arithmetic. The parameter 'a' adds minimal complexity since the method remains routine.
Spec4.03o Inverse 3x3 matrix

6 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r r } a & 1 & 0 \\ 1 & 2 & 1 \\ - 1 & 3 & 4 \end{array} \right)\), where \(a \neq 1\). Find \(\mathbf { C } ^ { - 1 }\).

Question 6:
AnswerMarks Guidance
AnswerMarks Guidance
M1Show correct expansion process for \(3\times3\) or multiplication of \(\mathbf{C}\) and adj\(\mathbf{C}\)
M1Correct evaluation of any \(2\times2\)
\(\det \mathbf{C} = \Delta = 5a-5\)A1 Obtain correct answer
M1Show correct process for adjoint entries
\(\frac{1}{\Delta}\begin{pmatrix} 5 & -4 & 1 \\ -5 & 4a & -a \\ 5 & -3a-1 & 2a-1 \end{pmatrix}\)A1 Obtain at least 4 correct entries in adjoint
A1Obtain completely correct adjoint
B1 [7]Divide their adjoint by their determinant
## Question 6:

| Answer | Marks | Guidance |
|--------|-------|----------|
| | M1 | Show correct expansion process for $3\times3$ or multiplication of $\mathbf{C}$ and adj$\mathbf{C}$ |
| | M1 | Correct evaluation of any $2\times2$ |
| $\det \mathbf{C} = \Delta = 5a-5$ | A1 | Obtain correct answer |
| | M1 | Show correct process for adjoint entries |
| $\frac{1}{\Delta}\begin{pmatrix} 5 & -4 & 1 \\ -5 & 4a & -a \\ 5 & -3a-1 & 2a-1 \end{pmatrix}$ | A1 | Obtain at least 4 correct entries in adjoint |
| | A1 | Obtain completely correct adjoint |
| | B1 **[7]** | Divide their adjoint by their determinant |

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6 The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { r r r } a & 1 & 0 \\ 1 & 2 & 1 \\ - 1 & 3 & 4 \end{array} \right)$, where $a \neq 1$. Find $\mathbf { C } ^ { - 1 }$.

\hfill \mbox{\textit{OCR FP1 2011 Q6 [7]}}