Matrices for Edexcel IGCSE Mathematics Specification B: Methods and Worked Examples

Matrices are one of the distinguishing features of the Edexcel IGCSE Mathematics Specification B exam. While many IGCSE specifications stop at vectors, this course introduces matrix algebra and matrix transformations, giving students a first encounter with a tool that underpins university-level mathematics and applied sciences across Europe and beyond.

The edexcel igcse mathematics specification b matrices content is divided into two areas: Matrix Operations and Matrix Transformations. The first covers the arithmetic of matrices; the second applies that arithmetic to geometric transformations of the plane. Together, they form a self-contained topic that connects elegantly to the Vectors and Transformation Geometry section elsewhere in the specification.

Matrix operations

What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. The order (or size) of a matrix is stated as rows by columns. A 2 x 3 matrix has 2 rows and 3 columns. The specification limits matrices to order not more than 3 x 3.

Addition and subtraction

Two matrices can be added or subtracted only if they have the same order. The operation is performed element by element.

Worked example:

A = [3, 1; -2, 5] and B = [1, -4; 0, 2].

A + B = [3+1, 1+(-4); -2+0, 5+2] = [4, -3; -2, 7].

Scalar multiplication

Multiplying a matrix by a scalar means multiplying every element by that number.

3A = [9, 3; -6, 15].

Matrix multiplication

Two matrices can be multiplied if the number of columns in the first equals the number of rows in the second. For an m x n matrix multiplied by an n x p matrix, the result is m x p.

Worked example: Find AB where A = [2, 3; 1, -1] and B = [4, 0; -2, 5].

Row 1 of A times column 1 of B: 2(4) + 3(-2) = 8 - 6 = 2.

Row 1 of A times column 2 of B: 2(0) + 3(5) = 0 + 15 = 15.

Row 2 of A times column 1 of B: 1(4) + (-1)(-2) = 4 + 2 = 6.

Row 2 of A times column 2 of B: 1(0) + (-1)(5) = 0 - 5 = -5.

AB = [2, 15; 6, -5].

The identity matrix and the zero matrix

The identity matrix I is the multiplicative identity: AI = IA = A. For 2 x 2 matrices:

I = [1, 0; 0, 1].

The zero (null) matrix has every element equal to zero. Adding the zero matrix to any matrix leaves it unchanged, just as adding zero to any number leaves it unchanged.

Determinants of 2 x 2 matrices

For a 2 x 2 matrix M = [a, b; c, d], the determinant is det(M) = ad - bc.

Worked example: Find the determinant of M = [5, 3; 2, 4].

det(M) = 5(4) - 3(2) = 20 - 6 = 14.

A matrix is non-singular (invertible) if its determinant is not zero. If det(M) = 0, the matrix is singular and has no inverse.

Inverses of 2 x 2 matrices

The inverse of M = [a, b; c, d] is:

M^-1 = (1/det(M)) x [d, -b; -c, a].

Worked example: Find the inverse of M = [5, 3; 2, 4].

det(M) = 14 (calculated above).

M^-1 = (1/14) x [4, -3; -2, 5] = [4/14, -3/14; -2/14, 5/14] = [2/7, -3/14; -1/7, 5/14].

Check: MM^-1 should equal I.

Row 1: 5(2/7) + 3(-1/7) = 10/7 - 3/7 = 7/7 = 1. And 5(-3/14) + 3(5/14) = -15/14 + 15/14 = 0.

Row 2: 2(2/7) + 4(-1/7) = 4/7 - 4/7 = 0. And 2(-3/14) + 4(5/14) = -6/14 + 20/14 = 14/14 = 1.

MM^-1 = [1, 0; 0, 1] = I. Confirmed.

Matrix transformations

A 2 x 2 matrix can represent a transformation of the plane. Multiplying a position vector by the transformation matrix gives the image position. The matrices edexcel igcse exam tests reflections, rotations about the origin, and enlargements with centre at the origin.

Common transformation matrices

Worked example: The point P has coordinates (3, 1). Find the image of P under a rotation of 90 degrees anticlockwise about the origin.

The transformation matrix is [0, -1; 1, 0].

Multiply: [0, -1; 1, 0] x [3; 1] = [0(3) + (-1)(1); 1(3) + 0(1)] = [-1; 3].

The image is (-1, 3).

Combining transformations

The igcse 4mb1 matrices content includes finding the matrix for a combined transformation. If transformation A is followed by transformation B, the combined matrix is BA (B times A, not A times B). The second transformation is on the left because it acts on the result of the first.

Worked example: Find the single matrix that represents a reflection in the y-axis followed by a rotation of 90 degrees anticlockwise about the origin.

Reflection in y-axis: R = [-1, 0; 0, 1].

Rotation 90 degrees anticlockwise: Q = [0, -1; 1, 0].

Combined = QR (rotation acts second, so it goes on the left).

QR = [0(-1) + (-1)(0), 0(0) + (-1)(1); 1(-1) + 0(0), 1(0) + 0(1)] = [0, -1; -1, 0].

This is the matrix for reflection in y = -x, which makes geometric sense: reflecting in the y-axis and then rotating 90 degrees anticlockwise is equivalent to reflecting in the line y = -x.

Solving systems with matrices

The inverse of a matrix can solve a system of two linear equations. If the system is Mx = b, where M is the coefficient matrix, x is the variable vector, and b is the constants vector, then x = M^-1b.

Worked example: Solve 3x + 2y = 7 and x - y = 1.

In matrix form: [3, 2; 1, -1] [x; y] = [7; 1].

det = 3(-1) - 2(1) = -5.

M^-1 = (-1/5) x [-1, -2; -1, 3] = [1/5, 2/5; 1/5, -3/5].

[x; y] = [1/5, 2/5; 1/5, -3/5] x [7; 1] = [7/5 + 2/5; 7/5 - 3/5] = [9/5; 4/5].

x = 9/5 = 1.8, y = 4/5 = 0.8.

Common mistakes in Matrices

• Order of multiplication in combined transformations. "A followed by B" means multiply BA, not AB. The second transformation goes on the left.
• Determinant sign errors. det = ad - bc, not ad + bc. The minus sign is critical; an incorrect determinant propagates into every element of the inverse.
• Swapping elements incorrectly in the inverse. In the inverse formula, swap a and d, then negate b and c. Forgetting to negate is a frequent error.
• Multiplying incompatible matrices. A 2 x 3 matrix cannot be multiplied by a 2 x 2 matrix (columns of the first must equal rows of the second).
• Confusing singular and non-singular. If det = 0, the matrix has no inverse. Attempting to compute one produces division by zero.

Self-check questions

Use these edexcel igcse mathematics specification b practice questions to test your understanding.

1. Given A = [2, -1; 3, 4] and B = [1, 5; -2, 0], find AB and BA. Are they equal?
2. Find the inverse of C = [4, 7; 3, 5] and verify by computing CC^-1.
3. A triangle has vertices at (1, 0), (0, 2), and (3, 1). Find the image of each vertex under the transformation represented by [0, 1; -1, 0]. What transformation is this?
4. Find the single matrix representing an enlargement of scale factor 2 (centre origin) followed by a reflection in the x-axis.

Solutions

1. AB = [2(1)+(-1)(-2), 2(5)+(-1)(0); 3(1)+4(-2), 3(5)+4(0)] = [4, 10; -5, 15]. BA = [1(2)+5(3), 1(-1)+5(4); -2(2)+0(3), -2(-1)+0(4)] = [17, 19; -4, 2]. AB is not equal to BA.

2. det(C) = 4(5) - 7(3) = 20 - 21 = -1. C^-1 = (1/(-1)) x [5, -7; -3, 4] = [-5, 7; 3, -4]. CC^-1 = [4(-5)+7(3), 4(7)+7(-4); 3(-5)+5(3), 3(7)+5(-4)] = [-20+21, 28-28; -15+15, 21-20] = [1, 0; 0, 1] = I.

3. [0,1;-1,0] x [1;0] = [0;-1]. [0,1;-1,0] x [0;2] = [2;0]. [0,1;-1,0] x [3;1] = [1;-3]. Images: (0,-1), (2,0), (1,-3). This is a rotation of 90 degrees clockwise about the origin.

4. Enlargement: E = [2,0;0,2]. Reflection in x-axis: R = [1,0;0,-1]. Combined = RE = [1,0;0,-1][2,0;0,2] = [2,0;0,-2].

The edexcel igcse mathematics specification b revision notes for Matrices should include a reference table of standard transformation matrices and the inverse formula. These are tools you will reach for repeatedly in the exam. The edexcel igcse mathematics specification b explained approach to matrices treats them as a precise, systematic method for handling transformations, and the edexcel igcse mathematics specification b notes you build from practice will serve as a compact reference for both papers.