A2 June 2023 Paper 1 Q6
6 The matrices \(\mathbf{M}\) and \(\mathbf{N}\) are \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) and \(\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\) respectively.
(a) In this question you must show detailed reasoning.
Determine whether \(\mathbf{M}\) and \(\mathbf{N}\) commute under matrix multiplication. [3]
Determine whether \(\mathbf{M}\) and \(\mathbf{N}\) commute under matrix multiplication. [3]
(b) Specify the transformation of the plane associated with each of the following matrices.
(i) \(\mathbf{M}\) [1]
(ii) \(\mathbf{N}\) [2]
(c) State the significance of the result in part (a) for the transformations associated with \(\mathbf{M}\) and \(\mathbf{N}\). [1]
(d) Use an algebraic method to show that all lines parallel to the \(x\)-axis are invariant lines of the transformation associated with \(\mathbf{N}\). [2]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\mathbf{MN} = \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix}\) or \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix}\) \(\mathbf{NM} = \begin{pmatrix} 0 & 2 \\ 1 & 0 \end{pmatrix}\) or \(\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ 1 & 0 \end{pmatrix}\) | M1 A1* | 1.1a 1.1 |
| so not commutative | A1 | 2.2a |
| [3] |
Notes
M1: both \(\mathbf{MN}\) and \(\mathbf{NM}\) calculated
A1*: both correct
A1: dep A1*
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\mathbf{M}\) is reflection in \(y = x\) | B1 | 1.1 |
| [1] | ||
| (ii) \(\mathbf{N}\) is stretch parallel to the \(x\)-axis | M1 | 1.1 |
| scale factor 2 | A1 | 1.1 |
| [2] |
| Scheme | Marks | AO |
|---|---|---|
| order of transformations matters | B1 | 2.2a |
| [1] |
Notes
B1: must refer to transformations not just matrices
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2x \\ y \end{pmatrix}\) or \(\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} x \\ mx + c \end{pmatrix} = \begin{pmatrix} 2x \\ mx + c \end{pmatrix}\) | M1 | 2.1 |
| \(y\)-coordinate unchanged so lines parallel to \(x\)-axis invariant | A1 | 2.2a |
| [2] |
Notes
M1: Allow SC1 for correct geometrical argument, e.g. stretch in \(x\)-direction leaves \(y\)-coordinates unchanged
A1: www. Must refer to invariant lines.