Matrix Multiplication: Introduction Revision Notes | GCSE AQA Higher Further Maths

1. Why Matrix Multiplication Matters

You already know how to add matrices and multiply a matrix by a scalar. Matrix multiplication takes things further — it lets you combine two transformations into one, or apply a transformation to a point in one clean calculation.

At GCSE, all matrix multiplication is restricted to 2x2 matrices multiplied by either another 2x2 matrix or a 2x1 matrix. That keeps the arithmetic manageable, but the method must be precise — one slip in the row-by-column process and every entry in your answer is wrong.

Think of a 2x1 matrix as a column vector representing a point, say $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$. Multiplying a 2x2 transformation matrix by that column vector tells you exactly where the point lands after the transformation. That connection between multiplication and geometry is what makes this topic so powerful.

2. The Row-by-Column Method: How It Works

The core idea is this: each entry in the product matrix comes from one row of the left matrix and one column of the right matrix. You multiply the corresponding entries together, then add them up.

For two 2x2 matrices $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$, the product $AB$ has four entries, each requiring its own row-by-column calculation.

A useful way to track which row and column you are using: cover the rest of the matrix with your fingers, leaving only the row you need from $A$ and the column you need from $B$. Then multiply entry by entry and add. This physical habit prevents the most common errors.

3. Multiplying Two 2x2 Matrices: Worked Example

Theory is clear on paper, but the row-by-column method only becomes automatic through practice. Here is a full walkthrough — follow each step and try to predict the next entry before revealing it.

Worked example: Multiplying Two 2x2 Matrices

Calculate $AB$ where $A = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 0 \\ 1 & 2 \end{pmatrix}$.

1. $AB = \begin{pmatrix} \text{(R1 of A)} \cdot \text{(C1 of B)} & \text{(R1 of A)} \cdot \text{(C2 of B)} \\ \text{(R2 of A)} \cdot \text{(C1 of B)} & \text{(R2 of A)} \cdot \text{(C2 of B)} \end{pmatrix}$ — Each of the four entries needs its own row-by-column calculation. [M1 for correct method structure]
2. Row 1 of $A$: $(3, 1)$ \quad Column 1 of $B$: $(5, 1)$ $(3 \times 5) + (1 \times 1) = 15 + 1 = 16$ — Multiply corresponding entries and add: $3 \times 5 = 15$, $1 \times 1 = 1$, total $= 16$.
3. Row 1 of $A$: $(3, 1)$ \quad Column 2 of $B$: $(0, 2)$ $(3 \times 0) + (1 \times 2) = 0 + 2 = 2$ — Same row 1 of $A$, but now paired with column 2 of $B$.
4. Row 2 of $A$: $(2, 4)$ \quad Column 1 of $B$: $(5, 1)$ $(2 \times 5) + (4 \times 1) = 10 + 4 = 14$ — Now switch to row 2 of $A$, paired with column 1 of $B$.
5. Row 2 of $A$: $(2, 4)$ \quad Column 2 of $B$: $(0, 2)$ $(2 \times 0) + (4 \times 2) = 0 + 8 = 8$ — Row 2 of $A$ paired with column 2 of $B$. [A1 for all four correct entries]
6. $AB = \begin{pmatrix} 16 & 2 \\ 14 & 8 \end{pmatrix}$ — Assemble the four entries in their correct positions.

Answer: $AB = \begin{pmatrix} 16 & 2 \\ 14 & 8 \end{pmatrix}$

4. Multiplying a 2x2 Matrix by a 2x1 Matrix

The second case you need to master is multiplying a 2x2 matrix by a 2x1 matrix. The 2x1 matrix has only one column, so the product has only one column too — it is itself a 2x1 matrix.

This is the calculation you use when a transformation matrix acts on a point. If the point is $\begin{pmatrix} x \\ y \end{pmatrix}$ and the transformation matrix is $M$, then $M\begin{pmatrix} x \\ y \end{pmatrix}$ gives the image point directly.

Worked example: Multiplying a 2x2 Matrix by a 2x1 Matrix

Calculate $M\mathbf{v}$ where $M = \begin{pmatrix} 2 & -1 \\ 3 & 5 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}$.

1. $M\mathbf{v} = \begin{pmatrix} 2 & -1 \\ 3 & 5 \end{pmatrix} \begin{pmatrix} 4 \\ 1 \end{pmatrix} = \begin{pmatrix}? \\? \end{pmatrix}$ — The result is a 2x1 matrix. There are two entries to find, one per row of $M$.
2. Row 1 of $M$: $(2, -1)$ \quad Column of $\mathbf{v}$: $(4, 1)$ $(2 \times 4) + (-1 \times 1) = 8 + (-1) = 7$ — $2 \times 4 = 8$ and $-1 \times 1 = -1$. Adding: $8 - 1 = 7$. [M1 for correct row-by-column method]
3. Row 2 of $M$: $(3, 5)$ \quad Column of $\mathbf{v}$: $(4, 1)$ $(3 \times 4) + (5 \times 1) = 12 + 5 = 17$ — $3 \times 4 = 12$ and $5 \times 1 = 5$. Adding: $12 + 5 = 17$. [A1 for both correct entries]
4. $M\mathbf{v} = \begin{pmatrix} 7 \\ 17 \end{pmatrix}$ — The image of the point $(4, 1)$ under transformation $M$ is the point $(7, 17)$.

Answer: $M\mathbf{v} = \begin{pmatrix} 7 \\ 17 \end{pmatrix}$

5. Formula Reference and Practice

Before moving to independent practice, here is a compact reference for both multiplication cases. Keep this in mind as you work through the problems below.

6. Exam Technique and Common Mistakes

Matrix multiplication questions are reliable mark-earners once the method is secure — but there are several traps that appear repeatedly in exam scripts. Knowing them in advance means you can sidestep them under pressure.

One of the most important things to remember is that matrix multiplication is not commutative. That means $AB$ and $BA$ are generally different matrices. The order you write the matrices matters — always check which matrix is on the left and which is on the right before you start.

Matrix multiplication — whether 2x2 by 2x2 or 2x2 by 2x1 — always follows the same row-by-column rule: pair the correct row with the correct column, multiply entry by entry, and add. Keeping that process systematic and showing full working is what secures every available mark.