AS June 2024 Paper 1 Q2
2 The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & a \\ -1 & 2 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & 0 \\ 1 & -1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} -1 & 0 \\ 2 & 1 \end{pmatrix}\), where \(a\) is a constant.
(a) By multiplying out the matrices on both sides of the equation, verify that \(\mathbf{A}(\mathbf{BC}) = (\mathbf{AB})\mathbf{C}\). [4]
(b) State the property of matrix multiplication illustrated by this result. [1]
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{A}(\mathbf{BC}) = \begin{pmatrix} 1 & a \\ -1 & 2 \end{pmatrix}\begin{pmatrix} -2 & 0 \\ -3 & -1 \end{pmatrix}\) | B1 | 1.1 |
| \(= \begin{pmatrix} -2 - 3a & -a \\ -4 & -2 \end{pmatrix}\) | B1 | 1.1 |
| \((\mathbf{AB})\mathbf{C} = \begin{pmatrix} 2 + a & -a \\ 0 & -2 \end{pmatrix}\begin{pmatrix} -1 & 0 \\ 2 & 1 \end{pmatrix}\) | B1 | 1.1 |
| \(= \begin{pmatrix} -2 - 3a & -a \\ -4 & -2 \end{pmatrix}\) [so \(\mathbf{AB}(\mathbf{C}) = \mathbf{A}(\mathbf{BC})\)] | B1 | 2.2a |
| [4] |
Notes
B1: \(\mathbf{BC} = \begin{pmatrix} -2 & 0 \\ -3 & -1 \end{pmatrix}\) (BC)
B1: (3rd) \(\mathbf{AB} = \begin{pmatrix} 2 + a & -a \\ 0 & -2 \end{pmatrix}\)
B1: (4th) cao
| Scheme | Marks | AO |
|---|---|---|
| Associativity | B1 | 1.2 |
| [1] |