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IB Mathematics: Applications and Interpretation HL · AHL 5: Matrices · AHL 5.8

Matrices as transformations Problem set A One question at a time

Try each question on paper first. Open the working a line at a time, then say whether you had it.

Skills

  1. Find the image of each point under the transformation with the given matrix.

    1. (a) \(\begin{pmatrix}2 & -3 \\ 1 & 4\end{pmatrix}\), point \((4, 1)\)

      • \(\text{Image} = M \times \text{position vector}\)
      • \(\begin{pmatrix}2 & -3 \\ 1 & 4\end{pmatrix}\begin{pmatrix}4 \\ 1\end{pmatrix} = \begin{pmatrix}5 \\ 8\end{pmatrix} = (5, 8)\)

    2. (b) \(\begin{pmatrix}1 & 2 \\ -1 & 3\end{pmatrix}\), point \((-2, 5)\)

      • \(\text{Image} = M \times \text{position vector}\)
      • \(\begin{pmatrix}1 & 2 \\ -1 & 3\end{pmatrix}\begin{pmatrix}-2 \\ 5\end{pmatrix} = \begin{pmatrix}8 \\ 17\end{pmatrix} = (8, 17)\)

    3. (c) \(\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\), point \((3, -7)\)

      • \(\text{Image} = M \times \text{position vector}\)
      • \(\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}3 \\ -7\end{pmatrix} = \begin{pmatrix}7 \\ 3\end{pmatrix} = (7, 3)\)

  2. Write down the matrix for each transformation.

    1. (a) Horizontal stretch, scale factor 5

      • \(\text{Image of } (1,0) \text{ is } (5,0) \text{ and image of } (0,1) \text{ is } (0,1).\)
      • \(\begin{pmatrix}5 & 0 \\ 0 & 1\end{pmatrix}\)

    2. (b) Vertical stretch, scale factor \(\frac{1}{3}\)

      • \(\text{Image of } (1,0) \text{ is } (1,0) \text{ and image of } (0,1) \text{ is } (0,\tfrac{1}{3}).\)
      • \(\begin{pmatrix}1 & 0 \\ 0 & \frac{1}{3}\end{pmatrix}\)

    3. (c) Enlargement, scale factor −2

      • \((1,0) \to (-2,0) \text{ and } (0,1) \to (0,-2).\)
      • \(\begin{pmatrix}-2 & 0 \\ 0 & -2\end{pmatrix}\)

    4. (d) Rotation 90° clockwise about the origin

      • \(\text{A 90° clockwise turn sends } (1,0) \text{ to } (0,-1) \text{ and } (0,1) \text{ to } (1,0).\)
      • \(\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}\)

    5. (e) Rotation 30° anticlockwise about the origin

      • \(\text{Rotation by } \theta \text{: } \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}, \; \theta = 30°.\)
      • \(\cos 30° = \tfrac{\sqrt{3}}{2},\; \sin 30° = \tfrac{1}{2}\)
      • \(\begin{pmatrix}\frac{\sqrt{3}}{2} & -0.5 \\ 0.5 & \frac{\sqrt{3}}{2}\end{pmatrix}\)

    6. (f) Reflection in the line \(y = \sqrt{3}\,x\)

      • \(\text{The line } y = \sqrt{3}\,x \text{ makes an angle } \theta = 60° \text{ with the } x\text{-axis} (\tan\theta = \sqrt{3}).\)
      • \(\text{Reflection: } \begin{pmatrix}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta\end{pmatrix}, \; 2\theta = 120°.\)
      • \(\cos 120° = -\tfrac{1}{2},\; \sin 120° = \tfrac{\sqrt{3}}{2}\)
      • \(\begin{pmatrix}-0.5 & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & 0.5\end{pmatrix}\)

  3. Find the point whose image is given.

    1. (a) \(\begin{pmatrix}2 & -3 \\ 1 & 4\end{pmatrix}\), image \((1, 6)\)

      • \(\text{Point} = M^{-1} \times \text{image.}\)
      • \(\det M = 11,\quad M^{-1} = \frac{1}{11}\begin{pmatrix}4 & 3 \\ -1 & 2\end{pmatrix}\)
      • \(\begin{pmatrix}\frac{4}{11} & \frac{3}{11} \\ -\frac{1}{11} & \frac{2}{11}\end{pmatrix}\begin{pmatrix}1 \\ 6\end{pmatrix} = \begin{pmatrix}2 \\ 1\end{pmatrix} = (2, 1)\)

    2. (b) \(\begin{pmatrix}1 & 2 \\ -1 & 3\end{pmatrix}\), image \((8, 7)\)

      • \(\text{Point} = M^{-1} \times \text{image.}\)
      • \(\det M = 5,\quad M^{-1} = \frac{1}{5}\begin{pmatrix}3 & -2 \\ 1 & 1\end{pmatrix}\)
      • \(\begin{pmatrix}0.6 & -0.4 \\ 0.2 & 0.2\end{pmatrix}\begin{pmatrix}8 \\ 7\end{pmatrix} = \begin{pmatrix}2 \\ 3\end{pmatrix} = (2, 3)\)

  4. The unit square has vertices \(O(0,0),\ A(1,0),\ C(1,1),\ B(0,1)\). Find the vertices of its image under each matrix, and sketch it.

    1. (a) \(\begin{pmatrix}3 & 1 \\ 1 & 2\end{pmatrix}\)

      • \(\text{O stays at } (0,0). \text{ The image of } (1,0) \text{ is the first column, and the image of } (0,1) \text{ is the second column.}\)
      • \(A \to (3, 1);\quad B \to (1, 2);\quad C = (1,1) \to (4, 3)\)
      • \(\text{Vertices: } O(0,0),\; (3,1),\; (4,3),\; (1,2)\)
      • Sketch the four vertices in order O, A, C, B and join them.

    2. (b) \(\begin{pmatrix}0 & -2 \\ 2 & 0\end{pmatrix}\)

      • \(\text{O stays at } (0,0). \text{ The image of } (1,0) \text{ is the first column, and the image of } (0,1) \text{ is the second column.}\)
      • \(A \to (0, 2);\quad B \to (-2, 0);\quad C = (1,1) \to (-2, 2)\)
      • \(\text{Vertices: } O(0,0),\; (0,2),\; (-2,2),\; (-2,0)\)
      • Sketch the four vertices in order O, A, C, B and join them.

    3. (c) \(\begin{pmatrix}1 & 2 \\ -1 & 1\end{pmatrix}\)

      • \(\text{O stays at } (0,0). \text{ The image of } (1,0) \text{ is the first column, and the image of } (0,1) \text{ is the second column.}\)
      • \(A \to (1, -1);\quad B \to (2, 1);\quad C = (1,1) \to (3, 0)\)
      • \(\text{Vertices: } O(0,0),\; (1,-1),\; (3,0),\; (2,1)\)
      • Sketch the four vertices in order O, A, C, B and join them.

Working backwards

  1. Each transformation sends \((1,0)\) and \((0,1)\) to the points given. Write down its matrix.

    1. (a) \((4, 2)\) and \((-1, 3)\)

      • \(\text{The images of } (1,0) \text{ and } (0,1) \text{ are the columns of the matrix.}\)
      • \(\begin{pmatrix}4 & -1 \\ 2 & 3\end{pmatrix}\)

    2. (b) \((0, 3)\) and \((-3, 0)\)

      • \(\text{The images of } (1,0) \text{ and } (0,1) \text{ are the columns of the matrix.}\)
      • \(\begin{pmatrix}0 & -3 \\ 3 & 0\end{pmatrix}\)

    3. (c) \((-1, 0)\) and \((2, 1)\)

      • \(\text{The images of } (1,0) \text{ and } (0,1) \text{ are the columns of the matrix.}\)
      • \(\begin{pmatrix}-1 & 2 \\ 0 & 1\end{pmatrix}\)

  2. Describe the transformation with each matrix.

    1. (a) \(\begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}\)

      • \((1,0) \to (0,-1),\; (0,1) \to (-1,0)\)
      • \(\text{Reflection in the line } y = -x \text{ swaps } (x,y) \to (-y,-x).\)

    2. (b) \(\begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}\)

      • \(\det = (-1)(-1) - 0 = 1;\; (1,0) \to (-1,0),\; (0,1) \to (0,-1).\)
      • \(\text{Every point goes to its opposite: a rotation of 180° about the origin (or an enlargement with scale factor } -1).\)

    3. (c) \(\begin{pmatrix}4 & 0 \\ 0 & 4\end{pmatrix}\)

      • \(\text{Both axes are scaled by 4, nothing else.}\)
      • An enlargement with scale factor 4, centre the origin.

    4. (d) \(\begin{pmatrix}0.6 & -0.8 \\ 0.8 & 0.6\end{pmatrix}\)

      • \(\text{Of the form } \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}: \cos\theta = 0.6,\; \sin\theta = 0.8.\)
      • \(\theta = \cos^{-1}(0.6) = 53.1°\)
      • A rotation of 53.1° anticlockwise about the origin.

Stretch

  1. (a) Find the matrix for a reflection in the line \(y = \frac{1}{3}x\). (b) Find the point whose image under this reflection is \((5, 5)\).

    • \(\text{(a) } \tan\theta = \tfrac{1}{3}: \text{ draw a right-angled triangle with sides 1, 3 and } \sqrt{10}, \text{ so } \cos\theta = \tfrac{3}{\sqrt{10}},\; \sin\theta = \tfrac{1}{\sqrt{10}}.\)
    • \(\cos 2\theta = \cos^2\theta - \sin^2\theta = \tfrac{9}{10} - \tfrac{1}{10} = 0.8,\quad \sin 2\theta = 2\sin\theta\cos\theta = \tfrac{6}{10} = 0.6\)
    • \(\begin{pmatrix}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta\end{pmatrix} = \begin{pmatrix}0.8 & 0.6 \\ 0.6 & -0.8\end{pmatrix}\)
    • \(\text{(b) Reflecting twice returns every point, so the original point is the reflection of } (5,5).\)
    • \(\begin{pmatrix}0.8 & 0.6 \\ 0.6 & -0.8\end{pmatrix}\begin{pmatrix}5 \\ 5\end{pmatrix} = \begin{pmatrix}7 \\ -1\end{pmatrix} = (7, -1)\)

jmaths.xyz · AI HL · AHL 5.8 Matrices as transformations