The Basics of a Matrix (VCE SSCE General Mathematics): Revision Notes

The Basics of a Matrix

What is a matrix?

A matrix is a rectangular arrangement of numbers organized in rows and columns. Matrices are incredibly useful for organizing and presenting data in a structured way.

Think of a matrix as a table that helps us keep track of related information. For example, imagine a market stall that operates on Fridays and Saturdays. We can record their sales of shirts, jeans, and belts in a matrix:

$A = \begin{bmatrix} 6 & 8 & 4 \\ 3 & 7 & 1 \end{bmatrix}$

To understand this matrix better, let's look at how rows and columns work:

Rows run horizontally (left to right):

• Row $1$ contains Friday's sales: $6, 8, 4$
• Row $2$ contains Saturday's sales: $3, 7, 1$

Columns run vertically (top to bottom):

• Column $1$ shows shirts sold: $6, 3$
• Column $2$ shows jeans sold: $8, 7$
• Column $3$ shows belts sold: $4, 1$

Order of a matrix

The order (or size) of a matrix tells us its dimensions. We express this as:

$\text{Order} = \text{number of rows} \times \text{number of columns}$

Examples of matrix orders

Let's look at matrices of different sizes:

• A matrix with $1$ row and $4$ columns has order 1 × 4
• A matrix with $3$ rows and $2$ columns has order 3 × 2
• A matrix with $4$ rows and $4$ columns has order 4 × 4

Elements of a matrix

The individual numbers within a matrix are called elements. Each element has a specific position that we can identify using subscript notation.

Element notation

We use lowercase letters with subscripts to identify elements. For a matrix called $A$, we write:

$a_{ij} = \text{the element in row } i, \text{ column } j$

The first subscript ($i$) tells us the row number, and the second subscript ($j$) tells us the column number.

Example: Consider the matrix:

$A = \begin{bmatrix} 6 & 8 & 4 \\ 3 & 7 & 1 \end{bmatrix}$

• Element $a_{13}$ is in row $1$, column $3$, and its value is 4
• Element $a_{22}$ is in row $2$, column $2$, and its value is 7
• Element $a_{21}$ is in row $2$, column $1$, and its value is 3

Worked example: Interpreting matrix elements

Types of matrices

Matrices come in different forms depending on their dimensions and properties. Let's explore the main types.

Row matrices

A row matrix consists of a single horizontal row of elements. Its order is always $1 \times n$, where $n$ is the number of columns.

Example: The Friday sales from our market stall can be written as:

$\begin{bmatrix} 6 & 8 & 4 \end{bmatrix}$

This is a 1 × 3 row matrix.

Column matrices

A column matrix consists of a single vertical column of elements. Its order is always $n \times 1$, where $n$ is the number of rows.

Example: The jeans sales from our market stall can be written as:

$\begin{bmatrix} 8 \\ 7 \end{bmatrix}$

This is a 2 × 1 column matrix.

Square matrices

In a square matrix, the number of rows equals the number of columns. These matrices have orders like $1 \times 1$, $2 \times 2$, $3 \times 3$, and so on.

Examples of square matrices:

$1 \times 1$ matrix: $\begin{bmatrix} 9 \end{bmatrix}$

$2 \times 2$ matrix: $\begin{bmatrix} 5 & 4 \\ 6 & 2 \end{bmatrix}$

$3 \times 3$ matrix: $\begin{bmatrix} 0 & 4 & 3 \\ 4 & 1 & 6 \\ 3 & 6 & 7 \end{bmatrix}$

Symmetric matrices

A symmetric matrix is a special type of square matrix where elements can be swapped according to this rule:

$a_{ij} = a_{ji}$

This means the element in row $i$, column $j$ equals the element in row $j$, column $i$.

Example: The $3 \times 3$ matrix shown above is symmetric because:

• $a_{12} = 4$ and $a_{21} = 4$ ✓
• $a_{13} = 3$ and $a_{31} = 3$ ✓
• $a_{23} = 6$ and $a_{32} = 6$ ✓

Working with matrices on calculators

Modern graphing calculators can store and manipulate matrices, making calculations much easier. Here's what you need to know:

Creating a matrix

To create a matrix on a calculator:

1. Access the matrix creation tool
2. Specify the number of rows
3. Specify the number of columns
4. Enter the values for each position

Displaying matrices

Once entered, calculators can display the complete matrix and allow you to:

• View specific elements
• Perform calculations
• Store matrices with variable names (like $A$, $B$, $C$)

To find a specific element, use notation like b[2,1] to access element $b_{21}$.

Remember!