What is a Matrix? (VCE SSCE General Mathematics): Revision Notes

What is a Matrix?

Introduction to matrices

A matrix (plural: matrices) is a rectangular arrangement of numbers or symbols organised in rows and columns. Think of it as a table of values enclosed in square brackets.

Matrices are useful for organising data in a structured way. For example, if we collect information about students' heights, weights, ages and pulse rates, we can arrange this data in a table:

We call this matrix $D$ (for data matrix). Matrices are typically named using capital letters such as $A$, $B$, $C$, and so on.

Rows and columns

Rows and columns are the fundamental building blocks of any matrix.

Rows are numbered from the top downwards: row $1$, row $2$, row $3$, etc.

Columns are numbered from left to right: column $1$, column $2$, column $3$, etc.

In the matrix shown above, Row 2 is highlighted in orange and Column 3 is highlighted in yellow.

Order of a matrix

The order (or size) of a matrix tells us its dimensions.

For example, matrix $D$ above has eight rows and four columns, so its order is $8 \times 4$ (read as "eight by four").

When we state the order as $8 \times 4$, this means:

• $8$ rows
• $4$ columns

Elements in a matrix

The individual numbers or values inside a matrix are called elements (or entries).

Types of matrices

Matrices come in many different shapes and sizes. Here are the main types you need to know:

Row matrices

A row matrix (also called a row vector) contains only one row of numbers.

Column matrices

A column matrix (also called a column vector) contains only one column of numbers.

The matrix $H$ shown above represents the heights of all students. It has order $8 \times 1$ (eight rows, one column) and contains $8$ elements.

Square matrices

A square matrix has an equal number of rows and columns.

Matrix $M$ shown above is a $4 \times 4$ square matrix containing data for male students. It has $4$ rows, $4$ columns, and $16$ elements in total.

Summary of basic matrix types

Diagonal matrices

Before understanding diagonal matrices, we need to identify the leading diagonal of a square matrix.

A square matrix has two diagonals, but the one running from the top-left to the bottom-right is called the leading diagonal. This is the most important diagonal in matrix mathematics.

Identity matrices

An identity matrix (also called a unit matrix) is a special type of diagonal matrix where each element on the leading diagonal equals $1$, and all other elements are $0$.

Identity matrices are denoted by the symbol $I$.

Every size of square matrix has its own identity matrix:

$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad I = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Symmetric matrices

A symmetric matrix is a square matrix that remains unchanged when we transpose it (swap its rows and columns).

In a symmetric matrix, the elements above the leading diagonal are a mirror image of the elements below it.

Triangular matrices

Triangular matrices come in two forms:

Upper triangular matrix: A square matrix where all elements below the leading diagonal are zeros.

Lower triangular matrix: A square matrix where all elements above the leading diagonal are zeros.

Transpose of a matrix

The transpose of a matrix is obtained by switching (interchanging) its rows and columns.

We denote the transpose of matrix $A$ by $A^T$.

Key points about transposition

For example: $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}^T = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

Matrix notation using subscripts

When we need to refer to individual elements within a matrix without specifying actual numbers, we use subscript notation.

For a matrix $A$, each element is identified by two subscripts:

• The first subscript indicates the row number
• The second subscript indicates the column number

So $a_{ij}$ represents the element in row $i$ and column $j$ of matrix $A$.

This notation is particularly useful when describing matrices in general terms or when working with matrix formulas.

Remember!