A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used extensively in VCE General Mathematics to represent data, solve systems of equations, and model real-world transitions and networks.
A matrix is denoted by a capital letter, such as $A$, $B$, or $T$. Its entries are referred to using subscript notation: $a_{ij}$ is the element in row $i$, column $j$.
$$A = \begin{pmatrix} 3 & 7 & 2 \ 1 & 5 & 9 \end{pmatrix}$$
This matrix $A$ has 2 rows and 3 columns, so its order is \$2 \times 3$ (read “2 by 3”).
| Type | Definition | Example |
|---|---|---|
| Row matrix | Only 1 row | $\begin{pmatrix} 4 & 2 & 1 \end{pmatrix}$ |
| Column matrix | Only 1 column | $\begin{pmatrix} 3 \ 7 \end{pmatrix}$ |
| Square matrix | Same number of rows and columns | $\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ |
| Zero matrix | All entries are 0 | $\begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}$ |
| Identity matrix | Square; 1s on main diagonal, 0s elsewhere | $I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$ |
The identity matrix $I$ acts like the number 1 in matrix multiplication: for any compatible matrix $A$,
$$AI = IA = A$$
For a \$3 \times 3$ identity:
$$I_3 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}$$
Matrices appear across multiple topics:
- Data storage: organising information in tables
- Transition problems: modelling how populations move between states
- Network adjacency: representing connections between nodes
- Financial modelling: applying matrix multiplication to allocation problems
A sports club records wins (W), losses (L), and draws (D) for two teams across a weekend:
$$\text{Results} = \begin{pmatrix} 3 & 1 & 0 \ 2 & 2 & 1 \end{pmatrix}$$
Row 1 = Team A: 3 wins, 1 loss, 0 draws. Row 2 = Team B: 2 wins, 2 losses, 1 draw.
The order of this matrix is \$2 \times 3$.
KEY TAKEAWAY: A matrix is defined by its order (rows $\times$ columns) and its entries $a_{ij}$. The identity matrix $I$ is the multiplicative identity for square matrices.
VCAA FOCUS: VCAA questions frequently ask you to state the order of a matrix, identify its type, or read off specific entries using subscript notation.
