| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Singular matrix conditions |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question testing basic matrix concepts: computing a 2×2 inverse using the standard formula and finding when determinant equals zero. Both parts require only direct application of standard procedures with minimal algebraic manipulation (solving 2a² - 18 = 0). While it's Further Maths content, the techniques are routine and mechanical. |
| Spec | 4.03l Singular/non-singular matrices4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(M = \begin{pmatrix} 4 & 3 \\ 6 & 2 \end{pmatrix}\) | B1 | Determinant must be \(-10\) |
| \(M^{-1} = \frac{1}{-10}\begin{pmatrix} 2 & -3 \\ -6 & 4 \end{pmatrix} = \begin{pmatrix} -0.2 & 0.3 \\ 0.6 & -0.4 \end{pmatrix}\) | M1 A1 | M1: for correct attempt at changing elements in major diagonal and changing signs in minor diagonal. Three or four of the numbers in the matrix should be correct – eg allow one slip. A1: for any form of correct answer, with correct determinant then isw. |
| (b) Setting \(\Delta = 0\) and using \(2a^2 \pm 18 = 0\) to obtain \(a = \pm 3\) | M1, A1 cao | (2 marks) |
**(a)** $M = \begin{pmatrix} 4 & 3 \\ 6 & 2 \end{pmatrix}$ | **B1** | Determinant must be $-10$
$M^{-1} = \frac{1}{-10}\begin{pmatrix} 2 & -3 \\ -6 & 4 \end{pmatrix} = \begin{pmatrix} -0.2 & 0.3 \\ 0.6 & -0.4 \end{pmatrix}$ | **M1 A1** | M1: for correct attempt at changing elements in major diagonal and changing signs in minor diagonal. Three or four of the numbers in the matrix should be correct – eg allow one slip. A1: for any form of correct answer, with correct determinant then isw.
**(b)** Setting $\Delta = 0$ and using $2a^2 \pm 18 = 0$ to obtain $a = \pm 3$ | **M1, A1 cao** | (2 marks)
**Guidance:** Special case: $a$ not replaced is **B0M1A0**
---2. $\mathbf { M } = \left( \begin{array} { c c } 2 a & 3 \\ 6 & a \end{array} \right)$, where $a$ is a real constant.
\begin{enumerate}[label=(\alph*)]
\item Given that $a = 2$, find $\mathbf { M } ^ { - 1 }$.
\item Find the values of $a$ for which $\mathbf { M }$ is singular.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2010 Q2 [5]}}