Matrices as transformations Problem set A Answers

1. (a) \((5, 8)\)
2. (b) \((8, 17)\)
3. (c) \((7, 3)\)

1. (a) \(\begin{pmatrix}5 & 0 \\ 0 & 1\end{pmatrix}\)
2. (b) \(\begin{pmatrix}1 & 0 \\ 0 & \frac{1}{3}\end{pmatrix}\)
3. (c) \(\begin{pmatrix}-2 & 0 \\ 0 & -2\end{pmatrix}\)
4. (d) \(\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}\)
5. (e) \(\begin{pmatrix}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2}\end{pmatrix}\)
6. (f) \(\begin{pmatrix}-\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2}\end{pmatrix}\)

1. (a) \((2, 1)\)
2. (b) \((2, 3)\)

1. (a) \(O,\ (3,1),\ (4,3),\ (1,2)\)
2. (b) \(O,\ (0,2),\ (-2,2),\ (-2,0)\)
3. (c) \(O,\ (1,-1),\ (3,0),\ (2,1)\)

1. (a) \(\begin{pmatrix}4 & -1 \\ 2 & 3\end{pmatrix}\)
2. (b) \(\begin{pmatrix}0 & -3 \\ 3 & 0\end{pmatrix}\)
3. (c) \(\begin{pmatrix}-1 & 2 \\ 0 & 1\end{pmatrix}\)

1. (a) Reflection in the line \(y = -x\)
2. (b) Rotation of 180° about the origin (or an enlargement with scale factor −1)
3. (c) Enlargement, scale factor 4, centre the origin
4. (d) Rotation of 53.1° anticlockwise about the origin

(a) \(\tan\theta = \frac{1}{3}\) gives \(\cos 2\theta = 0.8,\ \sin 2\theta = 0.6\): \(\begin{pmatrix}0.8 & 0.6 \\ 0.6 & -0.8\end{pmatrix}\). (b) A reflection undoes itself, so reflect \((5,5)\): \((7, -1)\).