Transformations using matrices

Q: How do I derive a transformation matrix if I don't remember it?

Find the image of the standard base vectors and under the transformation. The first column of the matrix is the image of , the second column is the image of .

Edexcel International A-Level Further Mathematics· FP1 Sections 6.1 to 6.4· 45 min read

1. Matrix Representation of 2D Linear Transformations★★☆☆☆⏱ 10 min

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📘 Definition

Linear transformation

A transformation that maps straight lines to straight lines, preserves the origin, and satisfies linearity: $M (a u + b v) = a M u + b M v$ for scalars $a, b$ and vectors $u, v$ .

Example:

A reflection across the x-axis is a linear transformation, while a translation (shifting a shape without rotating/stretching) is not linear because it does not preserve the origin.

All linear transformations centred at the origin can be represented as 2×2 matrices acting on column vectors. The general rule for finding any transformation matrix is to calculate the image of the standard base vectors $i = (10)$ and $j = (01)$ under the transformation: the first column of the matrix is the image of $i$ , and the second column is the image of $j$ .

📐 Worked Example

Derive the matrix representing reflection in the x-axis.

1
Find the image of $i = (10)$ : reflection over the x-axis leaves this point unchanged, so the image is $(10)$ .
2
Find the image of $j = (01)$ : reflection over the x-axis flips the sign of the y-coordinate, so the image is $(0 - 1)$ .
3
Construct the transformation matrix using these images as columns:
4
$\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$

Exam tip:

Always derive unknown transformation matrices using base vectors if you cannot recall them; this method works for all linear transformations centred at the origin and will earn you method marks even if your final matrix is incorrect.

2. Core Transformation Matrices★★★☆☆⏱ 12 min

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You need to be able to use the following core transformation matrices. Only the anticlockwise rotation and general line reflection matrices are provided in the formula book; all others must be derived using base vectors if you do not recall them.

Transformation	Matrix	Key Note
Reflection in x-axis	$(10 0 - 1)$	Derive via base vectors
Reflection in y-axis	$(- 1 0 01)$	Derive via base vectors
Reflection in $y = x$	$(0110)$	Derive via base vectors
Reflection in $y = - x$	$(0 - 1 - 1 0)$	Derive via base vectors
Anticlockwise rotation $θ$ about O	$(cos θ sin θ - sin θ cos θ)$	Given in formula book
Stretch parallel to x-axis, scale $k$	$(k 0 01)$	Derive via base vectors
Stretch parallel to y-axis, scale $k$	$(10 0 k)$	Derive via base vectors
Enlargement scale $k$ about O	$(k 0 0 k)$	Derive via base vectors

📐 Worked Example

Find the image of point $(2, 3)$ after an anticlockwise rotation of 90° about the origin.

1
Substitute $θ = 9 0^{\circ}$ into the standard rotation matrix: $cos 9 0^{\circ} = 0$ , $sin 9 0^{\circ} = 1$ , so the matrix is:
2
$\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$
3
Multiply the matrix by the column vector of the original point:
4
$\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}2 \\ 3\end{pmatrix} = \begin{pmatrix}0(2) + (-1)(3) \\ 1(2) + 0(3)\end{pmatrix} = \begin{pmatrix}-3 \\ 2\end{pmatrix}$
5
The image point is $(- 3, 2)$ .

Exam tip:

For angles that are multiples of 90°, verify your rotation result by plotting the point and rotating it physically on a coordinate grid to avoid sign errors.

3. Combined Transformations★★★★☆⏱ 15 min

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When you apply two transformations sequentially, the combined matrix is the product of the individual matrices, in reverse order of the transformation sequence. If you apply transformation B first, then transformation A, the combined matrix is $A B$ , because the vector is first multiplied by B, then by A: $A (B v) = (A B) v$ .

📐 Worked Example

A shape is first reflected in the line $y = x$ , then rotated anticlockwise by 180° about the origin. Find the combined transformation matrix.

1
Identify individual matrices: reflection in $y = x$ (first transformation, B) = $(0110)$ ; 180° anticlockwise rotation (second transformation, A) = $(- 1 0 0 - 1)$ .
2
Combine in reverse order: combined matrix = $A B$ .
3
Calculate the product:
4
$\begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$
5
This is the matrix for reflection in $y = - x$ , which is a valid check of the result.

Exam tip:

If you are unsure of the order, test your combined matrix on a simple point like (1,0) by applying the transformations sequentially and comparing the result to the matrix multiplication output.

4. Inverse Transformations and Determinant as Area Scale Factor★★★★☆⏱ 12 min

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📘 Definition

Area scale factor

The ratio of the area of the image of a shape to the area of the original shape, equal to $∣ det (M) ∣$ . The sign of $det (M)$ indicates orientation: positive for orientation preserved, negative for orientation reversed (e.g. reflections).

Example:

A reflection has $det (M) = - 1$ , so the area of the shape is unchanged, but its orientation is flipped.

An inverse transformation exists if and only if $det (M) \neq = 0$ , and is represented by $M^{- 1}$ , which reverses the effect of $M$ . For a sequence of transformations with combined matrix $A B$ , the inverse transformation is $(A B)^{- 1} = B^{- 1} A^{- 1}$ (reverse order of inverses).

📐 Worked Example

A transformation has matrix $M = (2003)$ . A triangle with area 4 cm² is transformed using M. (a) Find the area of the image triangle. (b) Find the matrix of the inverse transformation.

1
Calculate the determinant of M:
2
$\det(M) = (2)(3) - (0)(0) = 6$
3
Area scale factor = $∣ det (M) ∣ = 6$ , so image area = $4 \times 6 = 24$ cm².
4
Use the 2×2 inverse formula $M^{- 1} = \frac{1}{d e t ( M )} (d - c - b a)$ :
5
$M^{-1} = \frac{1}{6}\begin{pmatrix}3 & 0 \\ 0 & 2\end{pmatrix} = \begin{pmatrix}\frac{1}{2} & 0 \\ 0 & \frac{1}{3}\end{pmatrix}$

Exam tip:

You do not need to calculate the coordinates of all vertices of a shape to find its transformed area; simply multiply the original area by the absolute value of the determinant to save time.

5. Common Pitfalls

Wrong move:

Multiplying transformation matrices in the order of the verbal sequence (B then A = BA instead of AB)

Why:

Matrices act on column vectors from the left, so the later transformation must be applied after the first, so it goes on the left of the product.

Correct move:

If transformation B is applied first, then A, compute the product AB, not BA, and test with a sample point to confirm.

Wrong move:

Using clockwise rotation angles directly in the given anticlockwise rotation matrix

Why:

The formula book only provides the matrix for anticlockwise rotation about the origin.

Correct move:

For a clockwise rotation by $θ$ , use the anticlockwise matrix with $θ$ replaced by $- θ$ , giving $(cos θ - sin θ sin θ cos θ)$ .

Wrong move:

Forgetting to take the absolute value of the determinant when calculating area scale factor

Why:

A negative determinant only indicates orientation reversal, not negative area.

Correct move:

Multiply the original area by $∣ det (M) ∣$ to find the area of the image shape.

Wrong move:

Deriving transformation matrices using row vectors instead of column vectors

Why:

The Edexcel specification uses the column vector convention, where image = M * vector, not vector * M.

Correct move:

Use the image of column vectors $(10)$ and $(01)$ as columns of your transformation matrix, not rows.

Wrong move:

Assuming inverse transformations exist for all matrices

Why:

Transformations with determinant zero are singular, and collapse shapes to a line or point, so they cannot be reversed.

Correct move:

First check that $det (M) \neq = 0$ before calculating an inverse transformation matrix.

6. Quick Reference Cheatsheet

Transformation	Matrix	Key Property
Reflection in x-axis	$(10 0 - 1)$	$det = - 1$
Reflection in y-axis	$(- 1 0 01)$	$det = - 1$
Reflection in $y = x$	$(0110)$	$det = - 1$
Reflection in $y = - x$	$(0 - 1 - 1 0)$	$det = - 1$
Anticlockwise rotation $θ$	$(cos θ sin θ - sin θ cos θ)$	$det = 1$
Stretch x-axis scale $k$	$(k 0 01)$	$det = k$
Stretch y-axis scale $k$	$(10 0 k)$	$det = k$
Enlargement scale $k$	$(k 0 0 k)$	$det = k^{2}$
Combined B then A	$A B$	$det = det (A) det (B)$
Inverse transformation	$M^{- 1}$	$det = 1/ det (M)$

7. Frequently Asked

What order do I multiply matrices for combined transformations?

If you apply transformation B first, then transformation A, the combined matrix is AB, not BA. Matrices act on column vectors from the left, so the later transformation is placed on the left of the product.

How do I derive a transformation matrix if I don't remember it?

Find the image of the standard base vectors $(10)$ and $(01)$ under the transformation. The first column of the matrix is the image of $(1, 0)$ , the second column is the image of $(0, 1)$ .

What does a negative determinant mean?

The negative sign indicates the transformation reverses the orientation of the shape (e.g. reflections have determinant = -1). The absolute value of the determinant is the area scale factor, so you always use the positive value for area calculations.

Going deeper

official_documentEdexcel IAL FP1 SpecificationRefer to section 6 for transformation content
official_documentEdexcel IAL Further Maths Formula BookContains standard rotation and reflection matrix formulas

What's Next

Now that you have mastered 2D matrix transformations for FP1, you can apply these skills to past paper questions, which often combine this topic with matrix operations and coordinate geometry problems. Make sure you practice enough questions to avoid common order-of-multiplication errors, which cost many students marks each year. If you are studying higher further maths units, you will encounter matrix transformations again in FP3, where they are extended to 3 dimensions. You should also review matrix inverse and determinant calculation to ensure you can answer full 6-8 mark exam questions on combined and inverse transformations.