Transformations of the Unit Square Revision Notes | GCSE AQA Higher Further Maths
The unit square has vertices (0, 0), (1, 0), (1, 1) and (0, 1); multiply the matrix by each vertex to transform it. Free notes and 10 questions answered.
Updated 14 Mar 2026 · Revise Right Team
Transformations of the Unit Square is part of Transformations of the unit square in the x-y plane in GCSE AQA Higher Further Maths.
3 min
Reading time
5
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GCSE AQA Higher
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What this lesson covers
The unit square has vertices (0, 0), (1, 0), (1, 1) and (0, 1); multiply the matrix by each vertex to transform it. Free notes and 10 questions answered.
Identify the unit square and its four vertices in the x-y plane
Represent rotations, reflections, and enlargements as 2x2 matrices
Apply a 2x2 transformation matrix to the unit square to find the image vertices
Recall the standard matrices for rotations of 90°, 180°, and 270° about the origin, reflections in x=0, y=0, y=x, y=-x, and enlargements centred on the origin
Key ideas to keep in view
Use this lesson to stay inside Transformations of the unit square in the x-y plane while connecting the detail back to the wider Matrix Transformations topic.
Matrix Transformations
Transformations of the unit square in the x-y plane
GCSE AQA Higher Further Maths
Revision notes
Read the core explanations from Transformations of the Unit Square before testing yourself from memory.
1. Why the Unit Square?
Matrices can encode geometric transformations — rotations, reflections, enlargements — as a single grid of four numbers. The unit square is the perfect test shape because its vertices sit at coordinates that make the arithmetic clean and the geometry visible.
Once you know how a matrix moves the unit square, you know exactly what that matrix does to any shape. That is the power of this approach, and it is why exam questions almost always start with the unit square.
2. Representing Transformations as 2x2 Matrices
A 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ transforms a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ by matrix multiplication. The result is a new column vector giving the image coordinates.
There is a shortcut worth knowing: the first column of the matrix tells you where A(1,0) maps to, and the second column tells you where C(0,1) maps to. The origin O(0,0) always maps to itself for these transformations. B(1,1) maps to the sum of the two columns.
3. The Standard Transformation Matrices
The specification requires you to know the matrices for specific rotations, reflections, and enlargements. Rather than memorising them as isolated facts, notice the pattern: each matrix is built by asking where A(1,0) and C(0,1) land after the transformation.
A reliable way to reconstruct any of these matrices from scratch: sketch the transformation on a small diagram, track where (1,0) and (0,1) land, then write those image coordinates as the columns of your matrix — first image as column 1, second image as column 2.
4. Applying a Matrix to the Unit Square — Worked Examples
Theory is straightforward; exam questions are where the method gets tested. The process is always the same: multiply the matrix by each vertex column vector in turn, then plot or state the image vertices. Work through the following examples step by step.
Worked example: Rotation 90° Anticlockwise About the Origin
The unit square has vertices O(0,0), A(1,0), B(1,1), C(0,1). Apply the matrix $\mathbf{M} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and state the image vertices.
$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0\times0 + (-1)\times0 \\ 1\times0 + 0\times0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ — The origin always maps to itself for transformations centred on the origin. [M1 — correct method shown]
$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0\times1 + (-1)\times0 \\ 1\times1 + 0\times0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ — A(1,0) maps to A'(0,1). Notice this is just reading the first column of M. [A1]
$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0\times1 + (-1)\times1 \\ 1\times1 + 0\times1 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$ — B(1,1) maps to B'(-1,1). This equals the sum of the two columns of M. [A1]
$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0\times0 + (-1)\times1 \\ 1\times0 + 0\times1 \end{pmatrix} = \begin{pmatrix} -1 \\ 0 \end{pmatrix}$ — C(0,1) maps to C'(-1,0). This is just reading the second column of M. [A1]
$O'(0,0),\quad A'(0,1),\quad B'(-1,1),\quad C'(-1,0)$ — The image square sits in the second quadrant, which is consistent with a 90° anticlockwise rotation.
Apply the matrix $\mathbf{M} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ to the unit square. State the image vertices and describe the transformation.
$(x,y) \rightarrow (-y,-x)$: coordinates swap and both negate. — This is a reflection in the line $y = -x$. The line $y=-x$ passes through B'(-1,-1) and the origin, confirming the mirror line.
Exam questions sometimes give you a matrix and ask you to name the transformation. The column-reading trick works in reverse: read column 1 to find where A(1,0) goes, read column 2 to find where C(0,1) goes, then match the pattern to the standard list.
Worked example: Identifying the Transformation from a Given Matrix
The matrix $\mathbf{M} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ is applied to the unit square. Identify the transformation fully.
Column 1 is $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$, so $A(1,0) \rightarrow A'(-1,0)$. — A moves from the positive x-axis to the negative x-axis.
Column 2 is $\begin{pmatrix} 0 \\ -1 \end{pmatrix}$, so $C(0,1) \rightarrow C'(0,-1)$. — C moves from the positive y-axis to the negative y-axis.
$(x,y) \rightarrow (-x,-y)$: both coordinates negate. — Every point is mapped through the origin to the diametrically opposite position.
Rotation of $180^\circ$ about the origin. — This matches the standard matrix for 180° rotation. State the angle and centre for full marks. [A1]
Answer: Rotation of $180^\circ$ about the origin.
6. Deliberate Practice
You have seen the method applied to rotations, reflections, and enlargements. Now test your fluency across all three transformation types before moving to exam technique.
7. Exam Technique and Common Mistakes
Most marks lost on this topic come from a small set of predictable errors. Knowing them in advance is half the battle.
One final check that is worth building into your routine: after applying a matrix, verify that the area of the image is consistent with the transformation. Rotations and reflections preserve area, so the image of the unit square should still have area 1. An enlargement with scale factor $k$ produces an image with area $k^2$. If your image area does not match, at least one vertex calculation contains an error.
With the standard matrices memorised and the column-reading shortcut in your toolkit, every question on transformations of the unit square reduces to the same five-step process above — making this one of the most reliable mark-scoring topics on the paper.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.
The unit square is the square with vertices O(0, 0), A(1, 0), B(1, 1) and C(0, 1). Its sides lie along the coordinate axes and each side has length 1, which makes it the ideal test shape for seeing what a transformation matrix does.
How do you transform the unit square with a matrix?
Multiply the matrix by each vertex column vector in turn. For the 90° anticlockwise rotation matrix with rows (0, −1) and (1, 0), the images are O'(0, 0), A'(0, 1), B'(−1, 1) and C'(−1, 0), so the image square sits in the second quadrant.
How can you read a transformation from the columns of a matrix?
The first column of a 2 × 2 transformation matrix gives the image of A(1, 0), and the second column gives the image of C(0, 1). The origin maps to itself and B(1, 1) maps to the sum of the two columns, so a matrix can be built or named by tracking where (1, 0) and (0, 1) land.
What matrix gives a reflection in the line y = −x?
The matrix with rows (0, −1) and (−1, 0) gives a reflection in the line y = −x, mapping (x, y) to (−y, −x): the coordinates swap and both negate. It sends A(1, 0) to (0, −1), B(1, 1) to (−1, −1) and C(0, 1) to (−1, 0).
What matrix gives a rotation of 180° about the origin?
The matrix with rows (−1, 0) and (0, −1) is a rotation of 180° about the origin. Its first column sends A(1, 0) to (−1, 0) and its second sends C(0, 1) to (0, −1), so (x, y) maps to (−x, −y).
What is the matrix for a rotation of 90° anticlockwise?
The matrix for a rotation of 90° anticlockwise about the origin has rows (0, −1) and (1, 0). Applied to the unit square, it maps A(1, 0) to (0, 1), B(1, 1) to (−1, 1) and C(0, 1) to (−1, 0), so the image sits in the second quadrant.
How do you identify a transformation from its matrix?
A transformation is named from its matrix by reading column 1 to find where A(1, 0) goes and column 2 to find where C(0, 1) goes, then matching the pattern. For the matrix with rows (−1, 0) and (0, −1), (x, y) maps to (−x, −y), a rotation of 180° about the origin.
How does an enlargement matrix change the area of the unit square?
An enlargement with scale factor k produces an image of the unit square with area k², while rotations and reflections preserve area, so the image still has area 1. Checking the image area is a useful way to catch an error in a vertex calculation.
How do you work out a transformation matrix from scratch?
A transformation matrix is worked out by sketching the transformation on a small diagram, tracking where (1, 0) and (0, 1) land, then writing those image coordinates as the columns of the matrix: the first image as column 1 and the second image as column 2.
What happens to the origin in a matrix transformation?
The origin O(0, 0) always maps to itself under a 2 × 2 matrix transformation of the unit square, because multiplying any such matrix by the zero column vector gives zero. This is why these rotations and enlargements are centred on the origin.