June 2018 Paper 2 Q19
19 Use matrix multiplication to show that, in the \(x\)–\(y\) plane,
- a reflection in the line \(\ y = -x\), followed by
- a rotation, \(90^\circ\) anticlockwise about the origin, followed by
- a reflection in the \(x\)-axis
is equivalent to a transformation by the identity matrix. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 C(BA) | ||
| \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) and \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) and indication that \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) is the identity matrix | B5 | for B5, products must be seen in correct order and results of products must be correct B4 a B5 response with no indication that \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) is the identity matrix B3 (reflection in \(y = -x\)) \(\ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) and (rotation) \(\ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) and (reflection in \(x\)-axis) \(\ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) B2 Any two of the above B1 Any one of the above |
| Alternative method 2 (CB)A | ||
| \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) and indication that \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) is the identity matrix | B5 | for B5, products must be seen in correct order and results of products must be correct B4 a B5 response with no indication that \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) is the identity matrix B3 (reflection in \(y = -x\)) \(\ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) and (rotation) \(\ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) and (reflection in \(x\)-axis) \(\ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) B2 Any two of the above B1 Any one of the above |
| Alternative method 3 transforms a general point | ||
| \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -y \\ -x \end{pmatrix}\) and \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} -y \\ -x \end{pmatrix} = \begin{pmatrix} x \\ -y \end{pmatrix}\) and \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} x \\ -y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\) and indication that \(\begin{pmatrix} x \\ y \end{pmatrix}\) has mapped to itself | B5 | for B5, products must be seen in correct order and results of products must be correct B4 a B5 response with no indication that \(\begin{pmatrix} x \\ y \end{pmatrix}\) has mapped to itself B3 (reflection in \(y = -x\)) \(\ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) and (rotation) \(\ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) and (reflection in \(x\)-axis) \(\ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) B2 Any two of the above B1 Any one of the above |
| Alternative method 4 transforms the unit square | ||
| \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\) \(= \begin{pmatrix} 0 & 0 & -1 & -1 \\ 0 & -1 & -1 & 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 0 & -1 & -1 \\ 0 & -1 & -1 & 0 \end{pmatrix}\) \(= \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & -1 & -1 \end{pmatrix}\) and \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & -1 & -1 \end{pmatrix}\) \(= \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\) and indication that unit square has mapped to itself | B5 | columns in 2 by 4 matrices can be in any order for B5, products must be seen in correct order and results of products must be correct B4 a B5 response with no indication that unit square has mapped to itself B3 (reflection in \(y = -x\)) \(\ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) and (rotation) \(\ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) and (reflection in \(x\)-axis) \(\ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) B2 Any two of the above B1 Any one of the above |
Additional guidance
| For B3, B2 and B1 the matrices must not be the answers to a product | |
| Must use matrix multiplication | |
| Ignore commas and lines within matrices | |
| Allow missing brackets if arrays are correct | |
| Examples of indication Alt 1 or 2 \(\ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \mathrm{I}\ \) or \(\ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\ \) which is the identity matrix Alt 3 \(\ \begin{pmatrix} x \\ y \end{pmatrix}\ \) which is the same as the original Alt 4 \(\ \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\ \) which is the same as the original | |
| Alts 1 and 2 Indications may be seen at the start of the solution | |
| Alt 3 \(\ \begin{pmatrix} x \\ y \end{pmatrix}\ \) must be algebraic |