Introduction to Matrices and Matrix Operations

This lesson introduces matrices — rectangular arrays of numbers that are used to represent and solve systems of equations, describe geometric transformations, and model a wide range of mathematical and real-world problems. Matrices are a central topic in AQA A-Level Further Mathematics.

What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. The order (or size) of a matrix with $m$ rows and $n$ columns is $m \times n$ .

A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}

This is a $2 \times 3$ matrix (2 rows, 3 columns).

The entry in row $i$ , column $j$ is denoted $a_{ij}$ . For the matrix above, $a_{12} = 2$ and $a_{23} = 6$ .

Special Matrices

Matrix type	Description	Example
Square matrix	Same number of rows and columns ( $n \times n$ )	$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$
Identity matrix $I$	Square matrix with 1s on the diagonal and 0s elsewhere	$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
Zero matrix $O$	All entries are 0	$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
Diagonal matrix	Square; non-zero entries only on the main diagonal	$\begin{pmatrix} 3 & 0 \\ 0 & 7 \end{pmatrix}$
Symmetric matrix	$A = A^T$ (entry $a_{ij} = a_{ji}$ )	$\begin{pmatrix} 1 & 5 \\ 5 & 2 \end{pmatrix}$
Column vector	$n \times 1$ matrix	$\begin{pmatrix} 3 \\ -1 \end{pmatrix}$
Row vector	$1 \times n$ matrix	$\begin{pmatrix} 2 & 5 \end{pmatrix}$

Matrix Addition and Subtraction

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

\begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} a+e & b+f \\ c+g & d+h \end{pmatrix}

Worked Example: Compute $\begin{pmatrix} 3 & 1 \\ -2 & 4 \end{pmatrix} + \begin{pmatrix} 0 & -1 \\ 5 & 2 \end{pmatrix}$ .

= \begin{pmatrix} 3 & 0 \\ 3 & 6 \end{pmatrix}

Scalar Multiplication

Multiplying a matrix by a scalar $k$ multiplies every entry by $k$ :

k\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ka & kb \\ kc & kd \end{pmatrix}

Example: $3\begin{pmatrix} 2 & -1 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} 6 & -3 \\ 0 & 12 \end{pmatrix}$

The Transpose

The transpose of a matrix $A$ , written $A^T$ , is obtained by swapping rows and columns:

If $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ , then $A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$

Properties:

$(A^T)^T = A$
$(A + B)^T = A^T + B^T$
$(kA)^T = kA^T$
$(AB)^T = B^T A^T$ (note the reversal of order)

Equality of Matrices

Two matrices $A$ and $B$ are equal if and only if:

They have the same order
Corresponding entries are equal: $a_{ij} = b_{ij}$ for all $i, j$

This is analogous to equating real and imaginary parts of complex numbers — a powerful technique for finding unknowns.

Worked Example: Find $x$ and $y$ if $\begin{pmatrix} 2x & 3 \\ y+1 & 5 \end{pmatrix} = \begin{pmatrix} 6 & 3 \\ 4 & 5 \end{pmatrix}$ .

$2x = 6 \Rightarrow x = 3$ ; $y + 1 = 4 \Rightarrow y = 3$ .

Properties of Matrix Addition

Property	Statement
Commutative	$A + B = B + A$
Associative	$(A + B) + C = A + (B + C)$
Additive identity	$A + O = A$
Additive inverse	$A + (-A) = O$

Practice Problems

Given $A = \begin{pmatrix} 1 & -2 \\ 3 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 1 \\ -1 & 5 \end{pmatrix}$ , compute $A + B$ , $A - B$ , and $3A$ .
Find the transpose of $\begin{pmatrix} 2 & 5 \\ -1 & 3 \\ 0 & 7 \end{pmatrix}$ .
Find $a$ and $b$ if $\begin{pmatrix} a + 1 & 2b \\ 3 & a - b \end{pmatrix} = \begin{pmatrix} 4 & 6 \\ 3 & 0 \end{pmatrix}$ .

Solutions:

$A + B = \begin{pmatrix} 5 & -1 \\ 2 & 5 \end{pmatrix}$ , $A - B = \begin{pmatrix} -3 & -3 \\ 4 & -5 \end{pmatrix}$ , $3A = \begin{pmatrix} 3 & -6 \\ 9 & 0 \end{pmatrix}$
$\begin{pmatrix} 2 & -1 & 0 \\ 5 & 3 & 7 \end{pmatrix}$
$a + 1 = 4$ gives $a = 3$ ; $2b = 6$ gives $b = 3$ . Check: $a - b = 0$ ✓

Summary

A matrix is a rectangular array; its order is rows × columns.
Addition/subtraction: same-order matrices, element by element.
Scalar multiplication: multiply every entry by the scalar.
The transpose swaps rows and columns.
The identity matrix $I$ and zero matrix $O$ play roles analogous to 1 and 0 in ordinary arithmetic.

Exam Tip: State the order of matrices clearly. Matrix operations require compatible dimensions — always check before proceeding.