| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Determinant calculation and singularity |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question requiring only routine calculation of a 3×3 determinant using cofactor expansion or the rule of Sarrus, followed by recalling the definition of singular/non-singular matrices. While 3×3 determinants are beyond standard A-level, this is a standard textbook exercise for FP1 with no problem-solving element. |
| Spec | 4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} - 1\begin{bmatrix} 1 & 1 \\ 1 & 3 \end{bmatrix} + 3\begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}\) | M1 | Show correct expansion process, allow sign slips |
| \(\frac{2 \times 5 - 1 \times 2 + 3 \times (-1)}{5}\) | A1 A1 B1ft | Obtain correct (unsimplified) expression; Obtain correct answer; State that M is non-singular as det M non-zero, ft their determinant |
$2\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} - 1\begin{bmatrix} 1 & 1 \\ 1 & 3 \end{bmatrix} + 3\begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}$ | M1 | Show correct expansion process, allow sign slips
$\frac{2 \times 5 - 1 \times 2 + 3 \times (-1)}{5}$ | A1 A1 B1ft | Obtain correct (unsimplified) expression; Obtain correct answer; State that M is non-singular as det M non-zero, ft their determinant
**Total: 4 marks**3 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l l } 2 & 1 & 3 \\ 1 & 2 & 1 \\ 1 & 1 & 3 \end{array} \right)$.\\
(i) Find the value of the determinant of $\mathbf { M }$.\\
(ii) State, giving a brief reason, whether $\mathbf { M }$ is singular or non-singular.
\hfill \mbox{\textit{OCR FP1 2006 Q3 [4]}}