Matrices as transformations Problem set B Answers

1. (a) \((3, 19)\)
2. (b) \((-4, 17)\)
3. (c) \((2, 6)\)

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

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

1. (a) \(\begin{pmatrix}5 & 2 \\ -1 & 3\end{pmatrix}\)
2. (b) \(\begin{pmatrix}0 & 4 \\ -4 & 0\end{pmatrix}\)
3. (c) \(\begin{pmatrix}2 & -3 \\ 0 & 1\end{pmatrix}\)

1. (a) Reflection in the line \(y = x\)
2. (b) Enlargement, scale factor −3, centre the origin
3. (c) Enlargement, scale factor 5, centre the origin
4. (d) Rotation of 36.9° anticlockwise about the origin

(a) \(\tan\theta = 2\) gives \(\cos 2\theta = -0.6,\ \sin 2\theta = 0.8\): \(\begin{pmatrix}-0.6 & 0.8 \\ 0.8 & 0.6\end{pmatrix}\). (b) A reflection undoes itself, so reflect \((10,5)\): \((-2, 11)\).