A2 October 2020 Paper 2 Q7
7 The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 0.6 & 2.4 \\ -0.8 & 1.8 \end{pmatrix}\).
(a) Find \(\det\mathbf{A}\). [1]
The matrix \(\mathbf{A}\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
(b) By considering the determinants of these transformations, determine the scale factor of the stretch. [2]
(c) Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer. [1]
(d) Find the angle of rotation. [2]
| Scheme | Marks | AO |
|---|---|---|
| \(\det\mathbf{A}\ (= 0.6 \times 1.8 - -0.8 \times 2.4) = 3\) | B1 | 1.1 |
| [1] |
| Scheme | Marks | AO |
|---|---|---|
| Determinant of rotation \(= 1\) | B1 | 1.1 |
| Determinant of rotation \(\times\) determinant of stretch \(= 1 \times \text{sf} = 3 \Rightarrow \text{sf} = 3\) | B1 | 2.2a |
| [2] |
| Scheme | Marks | AO |
|---|---|---|
| Since the second column of \(\mathbf{A}\) contains entries bigger than 1 (in magnitude) the stretch must be parallel to the \(y\)-axis. | B1 | 2.4 |
| [1] |
Notes
B1: Or any correct, complete explanation.
May see \(\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} \cos\theta & -3\sin\theta \\ \sin\theta & 3\cos\theta \end{pmatrix}\) or similar
| Scheme | Marks | AO |
|---|---|---|
| \(\sin\theta = -0.8\) and \(\cos\theta = 0.6\) oe | M1 | 2.2a |
| awrt \(-53^\circ\) (or \(-0.93\) rads) | A1 | 1.1 |
| [2] |
Notes
M1: Condone if only one equation
A1: or \(53^\circ\) (0.93 rads) clockwise or \(307^\circ\) (5.36 rads) (anticlockwise).