What is matrix algebra?

For 2 × 2 matrices the product is $pmatrix a & b \\ c & d pmatrixpmatrix e & f \\ g & h pmatrix = pmatrix ae + bg & af + bh \\ ce + dg & cf + dh pmatrix.$

Find pmatrix 4 & 2 \\ 3 & 1 pmatrix. [1 mark]

Show that pmatrix 2 & 4 \\ 1 & 2 pmatrix is singular. [2 marks]

CCEA CCEA AS 2020-style practice: The matrix is A = pmatrix 3 & 1 \\ 5 & 2 pmatrix. Find A and A^-1, and hence solve the simultaneous equations 3x + y = 4, 5x + 2y = 9.

The determinant is A = (3)(2) - (1)(5) = 6 - 5 = 1. The inverse swaps the leading diagonal, negates the other diagonal and divides by the determinant: A^-1 = 11pmatrix 2 & -1 \\ -5 & 3 pmatrix = pmatrix 2 & -1 \\ -5 & 3 pmatrix. Writing the equations as Apmatrix x \\ y pmatrix = pmatrix 4 \\ 9 pmatrix gives pmatrix x \\ y pmatrix = A^-1pmatrix 4 \\ 9 pmatrix = pmatrix 2(4) - 1(9) \\ -5(4) + 3(9) pmatrix = pmatrix -1 \\ 7 pmatrix. So x = -1, y = 7. Markers reward the determinant, the correct inverse formula, and using it to solve the system.

CCEA CCEA AS 2018-style practice: The matrix R represents an anticlockwise rotation of 90^ about the origin. Write down R, and find the image of the point (2, -3) under this rotation.

An anticlockwise rotation through angle θ about the origin has matrix pmatrix cosθ & -sinθ \\ sinθ & cosθ pmatrix. For θ = 90^, cos 90^ = 0 and sin 90^ = 1, so R = pmatrix 0 & -1 \\ 1 & 0 pmatrix. The image of (2, -3) is pmatrix 0 & -1 \\ 1 & 0 pmatrixpmatrix 2 \\ -3 pmatrix = pmatrix 0(2) + (-1)(-3) \\ 1(2) + 0(-3) pmatrix = pmatrix 3 \\ 2 pmatrix. So the image is (3, 2). Markers reward the correct rotation matrix and the matrix-vector multiplication.

Northern IrelandFurther MathsSyllabus dot point

How do matrices encode linear transformations, and how do you invert a 2x2 matrix?

Matrix algebra including addition, multiplication and the identity, the determinant and inverse of a 2x2 matrix, and matrices as linear transformations of the plane including rotations, reflections and enlargements.

A CCEA AS Further Maths answer on matrix addition and multiplication, the identity matrix, the determinant and inverse of a 2x2 matrix, singular matrices, and how matrices represent rotations, reflections and enlargements of the plane.

Generated by Claude Opus 4.812 min answerUpdated 2026-06-16

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking

CCEA wants you to handle matrix algebra (addition, scalar multiples, matrix products and the identity), find the determinant and inverse of a $2 \times 2$ matrix, recognise a singular matrix, and use matrices to represent transformations of the plane such as rotations, reflections and enlargements. Using an inverse to solve simultaneous equations is a standard exam task.

The answer

Matrix algebra

For $2 \times 2$ matrices the product is

\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix}.

The identity matrix

The determinant and inverse of a 2x2 matrix

Matrices as transformations

Key fact

A $2 \times 2$ matrix maps the point $(x, y)$ to $\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix}$ . Standard matrices include:

\text{rotation } \theta \text{ anticlockwise: } \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}, \qquad \text{enlargement scale } k: \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}.

A reflection in the

x

-axis is

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

and in the line

y = x

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

. The columns of the matrix are the images of

(1, 0)

and

(0, 1)

A combined transformation is the product of the matrices, applied right to left: $\mathbf{B}$ then $\mathbf{A}$ is $\mathbf{AB}$ .

Combining two transformations

Find the single matrix for a reflection in the $x$ -axis followed by an anticlockwise rotation of $90^\circ$ , and find the image of $(1, 2)$ .

step Write the two matrices

Reflection in the $x$ -axis: $\mathbf{M} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ . Rotation $90^\circ$ : $\mathbf{R} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ .

step Multiply in the correct order

Reflection first, then rotation, so the combined matrix is $\mathbf{R}\mathbf{M}$ :

\mathbf{R}\mathbf{M} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

step Apply to the point

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}.

The image is

(2, 1)

, which is the reflection in the line

y = x

Examples in context

Example 1. Computer graphics. Every rotation, scaling and reflection of an on-screen shape is a matrix multiplying the coordinates of its vertices. Combining moves into a single product matrix lets a graphics card transform thousands of points with one operation, exactly the "multiply the matrices" idea.

Example 2. Solving linear systems. Writing $\mathbf{A}\mathbf{x} = \mathbf{b}$ and computing $\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}$ is the matrix method for simultaneous equations. A singular matrix ( $\det = 0$ ) signals that the equations have no unique solution, which is why the determinant test matters.

Try this

Q1. Find $\det\begin{pmatrix} 4 & 2 \\ 3 & 1 \end{pmatrix}$ . [1 mark]

Cue. $\det = (4)(1) - (2)(3) = 4 - 6 = -2$ .

Q2. Write down the matrix for an enlargement, centre the origin, scale factor $3$ . [1 mark]

Cue. $\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$ .

Q3. Show that $\begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}$ is singular. [2 marks]

Cue. $\det = (2)(2) - (4)(1) = 4 - 4 = 0$ , so it has no inverse.

Exam-style practice questions

Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

CCEA AS 20206 marksThe matrix is

\mathbf{A} = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}

. Find

\det\mathbf{A}

and

\mathbf{A}^{-1}

, and hence solve the simultaneous equations

3x + y = 4

5x + 2y = 9

Show worked answer →

The determinant is $\det\mathbf{A} = (3)(2) - (1)(5) = 6 - 5 = 1.$

The inverse swaps the leading diagonal, negates the other diagonal and divides by the determinant:

$\mathbf{A}^{-1} = \dfrac{1}{1}\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}.$

Writing the equations as $\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 9 \end{pmatrix}$ gives

$\begin{pmatrix} x \\ y \end{pmatrix} = \mathbf{A}^{-1}\begin{pmatrix} 4 \\ 9 \end{pmatrix} = \begin{pmatrix} 2(4) - 1(9) \\ -5(4) + 3(9) \end{pmatrix} = \begin{pmatrix} -1 \\ 7 \end{pmatrix}.$

So $x = -1$ , $y = 7$ . Markers reward the determinant, the correct inverse formula, and using it to solve the system.

CCEA AS 20185 marksThe matrix

\mathbf{R}

represents an anticlockwise rotation of

90^\circ

about the origin. Write down

\mathbf{R}

, and find the image of the point

(2, -3)

under this rotation.

Show worked answer →

An anticlockwise rotation through angle $\theta$ about the origin has matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ .

For $\theta = 90^\circ$ , $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$ , so

$\mathbf{R} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$

The image of $(2, -3)$ is

$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 0(2) + (-1)(-3) \\ 1(2) + 0(-3) \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}.$

So the image is $(3, 2)$ . Markers reward the correct rotation matrix and the matrix-vector multiplication.

Related dot points

Sources & how we know this

CCEA GCE Further Mathematics specification — CCEA (2018)