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 \(j\).
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 \(A\),
For a \(3 \times 3\) identity:
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:
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.
