Edexcel IAL - Further Pure Mathematics 1- 6.2 2 × 2 Matrix Representations of Transformations- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 1- 6.2 2 × 2 Matrix Representations of Transformations -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 1- 6.2 2 × 2 Matrix Representations of Transformations -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 6.2 2 × 2 Matrix Representations of Transformations
Applications of \( 2 \times 2 \) Matrices in Geometrical Transformations
Geometrical transformations in the plane can be represented using \( 2 \times 2 \) matrices acting on column vectors.
A point \( (x,y) \) is written as the column vector:
\( \begin{pmatrix} x \\ y \end{pmatrix} \)
If a transformation is represented by the matrix \( A \), then the image of the point is:
\( A \begin{pmatrix} x \\ y \end{pmatrix} \)
All transformations considered here are about the origin \( (0,0) \).
1. Reflections
(a) Reflection in the x-axis
This changes the sign of the y-coordinate.
Matrix: \( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \)
(b) Reflection in the y-axis
This changes the sign of the x-coordinate.
Matrix: \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \)
(c) Reflection in the line \( y = x \)
This swaps the x- and y-coordinates.
Matrix: \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
(d) Reflection in the line \( y = -x \)
This swaps the coordinates and changes both signs.
Matrix: \( \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \)
2. Rotations About the Origin
A rotation through an angle \( \theta \) anticlockwise about \( (0,0) \) is represented by:
\( \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \)
Special cases often required:
- \( 90^\circ \): \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \)
- \( 180^\circ \): \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \)
- \( 270^\circ \): \( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \)
3. Stretches
(a) Stretch parallel to the x-axis
Scale factor \( k \) in the x-direction:
\( \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \)
(b) Stretch parallel to the y-axis
Scale factor \( k \) in the y-direction:
\( \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \)
4. Enlargement About the Origin
An enlargement with scale factor \( k \) about \( (0,0) \) multiplies both coordinates by \( k \).
Matrix: \( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \)
Here \( k \in \mathbb{R} \), \( k \ne 0 \). If \( k < 0 \), the image is also reflected in the origin.
Summary Table of Common Transformations
| Transformation | Matrix |
| Reflection in x-axis | \( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \) |
| Reflection in y-axis | \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \) |
| Reflection in \( y=x \) | \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \) |
| Rotation by \( \theta \) | \( \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \) |
| Enlargement (scale \( k \)) | \( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \) |
Example
Find the image of \( (3,-2) \) after reflection in the x-axis.
▶️ Answer / Explanation
\( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \)
Example
A point is rotated \( 90^\circ \) anticlockwise about the origin. Write down the transformation matrix.
▶️ Answer / Explanation
\( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \)
Example
A transformation consists of a reflection in the y-axis followed by an enlargement with scale factor 2. Find the combined transformation matrix.
▶️ Answer / Explanation
Reflection in y-axis: \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \)
Enlargement: \( \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \)
Combined matrix: \( \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 0 \\ 0 & 2 \end{pmatrix} \)