| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Matrix arithmetic operations |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question involving basic matrix operations (subtraction, multiplication) and solving for a parameter. Part (a) requires routine arithmetic, while part (b) involves equating to a scalar matrix and solving a simple equation. Despite being Further Maths content, the mechanical nature and standard techniques make it easier than average. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar |
The matrices $A$ and $B$ are defined by
$$A = \begin{pmatrix} 2 & 3 \\ 4 & p \end{pmatrix} \quad B = \begin{pmatrix} 1 & 3 \\ 2 & 3 \end{pmatrix}$$
**(a) Find, in terms of $p$, the matrices:**
(i) $A - B$ **(1 mark)**
(ii) $AB$ **(2 marks)**
**(b) Show that there is a value of $p$ for which $A - B + AB = kI$, where $k$ is an integer and $I$ is the $2 \times 2$ identity matrix, and state the corresponding value of $k$. (4 marks)**2 The matrices $\mathbf { A }$ and $\mathbf { B }$ are defined by
$$\mathbf { A } = \left[ \begin{array} { c c }
p & 2 \\
4 & p
\end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l }
3 & 1 \\
2 & 3
\end{array} \right]$$
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $p$, the matrices:
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { A } - \mathbf { B }$;
\item AB .
\end{enumerate}\item Show that there is a value of $p$ for which $\mathbf { A } - \mathbf { B } + \mathbf { A B } = k \mathbf { I }$, where $k$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix, and state the corresponding value of $k$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2013 Q2 [7]}}