| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Properties of matrix operations |
| Difficulty | Moderate -0.5 This is a straightforward Further Maths question testing basic matrix multiplication and properties. Parts (a) and (b) involve routine 2×2 matrix calculations, while part (c) demonstrates that matrix multiplication is non-commutative—a standard textbook exercise. The computations are simple and the concepts are fundamental to FP1, making this easier than average even for Further Maths. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(AB = \begin{bmatrix} 0 & -4 \\ 4 & 0 \end{bmatrix}\) | M1A1 | 2 marks |
| (b) \(A^2 = \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix}\); ... \(= 4I\) | B1, B1 | 2 marks |
| (c) \((AB)^2 = -16I\); \(B^2 = 4I\); so \(A^2 B^2 = 16I\) (hence result) | B1, B1, B1 | 3 marks |
**(a)** $AB = \begin{bmatrix} 0 & -4 \\ 4 & 0 \end{bmatrix}$ | M1A1 | 2 marks | M1A0 if 3 entries correct
**(b)** $A^2 = \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix}$; ... $= 4I$ | B1, B1 | 2 marks |
**(c)** $(AB)^2 = -16I$; $B^2 = 4I$; so $A^2 B^2 = 16I$ (hence result) | B1, B1, B1 | 3 marks | PI; Condone absence of conclusion
**Total: 7 marks**6 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by
$$\mathbf { A } = \left[ \begin{array} { l l }
0 & 2 \\
2 & 0
\end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { r r }
2 & 0 \\
0 & - 2
\end{array} \right]$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the matrix $\mathbf { A B }$.
\item Show that $\mathbf { A } ^ { 2 }$ is of the form $k \mathbf { I }$, where $k$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
\item Show that $( \mathbf { A B } ) ^ { 2 } \neq \mathbf { A } ^ { 2 } \mathbf { B } ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q6 [7]}}