Use matrices: add, subtract and multiply matrices, use the identity matrix, and apply 2x2 matrices to describe and combine transformations of the plane.

What this dot point is asking

A matrix is a rectangular array of numbers, and CCEA GCSE Further Mathematics uses $2 \times 2$ matrices to describe transformations of the plane. You must add, subtract and multiply matrices, know the identity matrix, and use a matrix to carry out and to identify transformations such as rotations, reflections and enlargements, including combining two transformations into one. This is distinctive Further Mathematics content not seen in ordinary GCSE.

Matrix arithmetic

Addition and subtraction work entry by entry and need matrices of the same size, so $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 5 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 7 \\ 4 & 5 \end{pmatrix}$. Scalar multiplication multiplies every entry by the number: $3\begin{pmatrix} 1 & 2 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 0 & 12 \end{pmatrix}$.

Multiplying matrices

Matrix multiplication is the key skill and follows the row-by-column rule: each entry of the product comes from multiplying the entries of a row of the first matrix by a column of the second and adding the results. Order matters, so in general $AB \ne BA$.

The identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ behaves like the number $1$: $AI = IA = A$ for any $2 \times 2$ matrix $A$, and it represents leaving the plane unchanged.

Matrices as transformations

A $2 \times 2$ matrix transforms a point by multiplying the point written as a column vector. To find the image of $(x, y)$, compute $\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$. The columns of the matrix are exactly the images of the unit vectors $(1, 0)$ and $(0, 1)$, which is why testing those two points identifies any transformation.

Combining transformations

Doing one transformation after another corresponds to multiplying their matrices, but in reverse order: to apply transformation $P$ first and then $Q$, the single matrix is $QP$, because $Q(P\mathbf{x}) = (QP)\mathbf{x}$. This is the same right-to-left reading as composite functions. So a rotation followed by a reflection is found by writing the reflection matrix on the left of the rotation matrix and multiplying. Always be careful with the order, because matrix multiplication is not commutative.

Why this matters

Matrices give a compact, powerful way to handle geometry, and they connect to the function and coordinate-geometry topics: a transformation is just a function on points, and combining them is matrix multiplication in place of composition. The row-by-column rule and the identity matrix are also the foundation for any further matrix work, so getting the arithmetic exact and the order right is what these questions reward.