Multiplying matrices Revision Notes for AQA GCSE Further Maths

Multiplying matrices

Understanding matrix multiplication

Matrix multiplication is a fundamental operation in linear algebra that allows us to combine matrices in meaningful ways. Unlike regular number multiplication, matrix multiplication follows specific rules and has unique properties that make it both powerful and sometimes tricky to master.

When we talk about multiplying matrices, we can encounter two main types: multiplying a matrix by a single number (called scalar multiplication) and multiplying one matrix by another matrix. Each type has its own method and rules.

Scalar multiplication

Scalar multiplication is the simpler of the two types. In this context, a scalar refers to a single number (not a matrix). When we multiply a matrix by a scalar, we multiply every single element in the matrix by that number.

The process is straightforward: take your scalar value and multiply it with each individual element in the matrix, keeping the same position for each result.

Example

Worked Example: Scalar Multiplication

When we have matrix $A = \begin{bmatrix} 2 & -3 \ -1 & 5 \end{bmatrix}$ and want to find $4A$ , we calculate:

$4\begin{bmatrix} 2 & -3 \ -1 & 5 \end{bmatrix} = \begin{bmatrix} 4 \times 2 & 4 \times (-3) \ 4 \times (-1) & 4 \times 5 \end{bmatrix} = \begin{bmatrix} 8 & -12 \ -4 & 20 \end{bmatrix}$

Notice how each element gets multiplied by 4, and the resulting matrix maintains the same dimensions as the original.

Matrix by matrix multiplication

Multiplying a 2×2 matrix by a 2×1 matrix

When multiplying matrices together, the process becomes more complex. For a 2×2 matrix multiplied by a 2×1 matrix, we multiply each row of the first matrix by the single column of the second matrix.

Here's how it works: take the first row of the 2×2 matrix and multiply corresponding elements with the column, then add these products together. Repeat this process for the second row.

Example

Worked Example: 2×2 by 2×1 Matrix Multiplication

Consider matrices $P = \begin{bmatrix} -3 & -1 \ 4 & 2 \end{bmatrix}$ and $Q = \begin{bmatrix} 6 \ 5 \end{bmatrix}$ . To find $PQ$ :

Step 1: First row calculation: $(-3) \times 6 + (-1) \times 5 = -18 + (-5) = -23$

Step 2: Second row calculation: $4 \times 6 + 2 \times 5 = 24 + 10 = 34$

Result: $PQ = \begin{bmatrix} -23 \ 34 \end{bmatrix}$

Multiplying a 2×2 matrix by a 2×2 matrix

For 2×2 by 2×2 multiplication, each row of the first matrix gets multiplied by each column of the second matrix. This creates a 2×2 result where each element comes from a specific row-column combination.

Example

Worked Example: 2×2 by 2×2 Matrix Multiplication

Take matrices $C = \begin{bmatrix} 0 & -1 \ -2 & 3 \end{bmatrix}$ and $D = \begin{bmatrix} 2 & 4 \ -3 & 1 \end{bmatrix}$ . To calculate $CD$ :

Element (1,1): $0 \times 2 + (-1) \times (-3) = 0 + 3 = 3$

Element (1,2): $0 \times 4 + (-1) \times 1 = 0 + (-1) = -1$

Element (2,1): $(-2) \times 2 + 3 \times (-3) = -4 + (-9) = -13$

Element (2,2): $(-2) \times 4 + 3 \times 1 = -8 + 3 = -5$

Result: $CD = \begin{bmatrix} 3 & -1 \ -13 & -5 \end{bmatrix}$

Rules and restrictions for matrix multiplication

Important

The Compatibility Rule

Matrices can only be multiplied when the number of columns in the first matrix equals the number of rows in the second matrix. This is called the compatibility condition.

For example, you cannot multiply $\begin{bmatrix} 3 \ 2 \end{bmatrix}$ by $\begin{bmatrix} -3 & -1 \ 4 & 2 \end{bmatrix}$ because the first matrix has 1 column but the second matrix has 2 rows.

When multiplication is possible, the resulting matrix will have dimensions equal to the number of rows from the first matrix and the number of columns from the second matrix.

Equating matrix elements

When two matrices are equal, their corresponding elements must also be equal. This principle becomes very useful when solving for unknown values in matrix equations.

Example

Worked Example: Solving for Unknown Elements

If we know that $\begin{bmatrix} 3 & a \ -2 & 1 \end{bmatrix}\begin{bmatrix} 2 & -2 \ -3 & b \end{bmatrix} = \begin{bmatrix} 12 & 2 \ -7 & 0 \end{bmatrix}$ , we can find the unknown values by:

Step 1: Calculate the left-hand side matrix multiplication

Step 2: Set corresponding elements equal to those on the right-hand side

Step 3: Solve the resulting equations

Following this method, we multiply the left side to get $\begin{bmatrix} 6-3a & -6+ab \ -7 & 4+b \end{bmatrix}$ , then equate elements to find $a = -2$ and $b = -4$ .

Properties of matrix multiplication

Note

Non-Commutative Property

One crucial property to understand is that matrix multiplication is not commutative. This means that $AB$ does not generally equal $BA$ , even when both multiplications are possible.

This non-commutative property makes matrix multiplication quite different from regular number multiplication and is something to always keep in mind when working with matrices.

Common exam tips

Note

When approaching matrix multiplication problems, always check the dimensions first to ensure multiplication is possible. Work systematically through each element calculation, and double-check your arithmetic since small errors can lead to completely wrong answers.

For scalar multiplication, remember that every element gets multiplied - don't miss any corners of the matrix. For matrix-by-matrix multiplication, take your time with the row-column combinations and consider writing out the intermediate steps to avoid confusion.

Remember!

Summary

Key Points to Remember:

Scalar multiplication: Multiply every element in the matrix by the scalar value
Compatibility rule: Number of columns in first matrix must equal number of rows in second matrix
Element calculation: Each element comes from multiplying a row by a column and adding the products
Non-commutative property: $AB \neq BA$ in general - order matters in matrix multiplication
Matrix equality: When matrices are equal, corresponding elements are equal - useful for solving unknowns

Multiplying matrices (AQA GCSE Further Maths): Revision Notes