| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Matrix arithmetic operations |
| Difficulty | Easy -1.2 This is a straightforward matrix arithmetic question requiring basic addition and multiplication of 2×2 matrices, followed by simple verification that a linear combination equals a scalar multiple of the identity matrix. All operations are routine calculations with no problem-solving or conceptual insight required, making it easier than average even for Further Maths. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| \(A + B = \begin{bmatrix} 3 & 6 \\ 6 & 3 \end{bmatrix}\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(AB = \begin{bmatrix} 8 & 6 \\ 6 & 8 \end{bmatrix}\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(A + B - AB = \begin{bmatrix} -5 & 0 \\ 0 & -5 \end{bmatrix}\) | B1F | — |
| \(\ldots = -5\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) | B1 | 2 marks |
**Part (a)(i)**
| $A + B = \begin{bmatrix} 3 & 6 \\ 6 & 3 \end{bmatrix}$ | M1A1 | 2 marks | M1A0 if 3 entries correct |
**Part (a)(ii)**
| $AB = \begin{bmatrix} 8 & 6 \\ 6 & 8 \end{bmatrix}$ | M1A1 | 2 marks | Ditto |
**Part (b)**
| $A + B - AB = \begin{bmatrix} -5 & 0 \\ 0 & -5 \end{bmatrix}$ | B1F | — | ft wrong answers in (a) |
| $\ldots = -5\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ | B1 | 2 marks | — |
**Total: 6 marks**
---1 The matrices $\mathbf { A }$ and $\mathbf { B }$ are defined by
$$\mathbf { A } = \left[ \begin{array} { l l }
3 & 4 \\
4 & 3
\end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l }
0 & 2 \\
2 & 0
\end{array} \right]$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the matrices:
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { A } + \mathbf { B }$;
\item $\mathbf { A B }$.
\end{enumerate}\item Show that $\mathbf { A } + \mathbf { B } - \mathbf { A B } = k \mathbf { I }$, where $k$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q1 [6]}}