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Home Subjects General Mathematics Matrix order & notation

Matrix Order and Notation

General Mathematics

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Area Of Study

Matrices

Unit

Unit 3: General Mathematics Unit 3

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Matrix Order and Notation

General Mathematics
01 May 2026

Matrix Order and Notation

Order of a Matrix

The order (or dimension) of a matrix is expressed as:

\[m \times n \quad (\text{rows} \times \text{columns})\]

A matrix with \(m\) rows and \(n\) columns has order \(m \times n\) and contains \(m \times n\) entries in total.

\[B = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \\ 0 & 1 & 2 \end{pmatrix} \quad \Rightarrow \quad \text{Order: } 4 \times 3\]

\(B\) has 4 rows and 3 columns, so 12 entries in total.

Element Notation

The element in row \(i\) and column \(j\) of matrix \(A\) is written \(a_{ij}\).

\[A = \begin{pmatrix} 3 & 8 & 1 \\ 6 & 2 & 9 \end{pmatrix}\]
  • \(a_{11} = 3\) (row 1, column 1)
  • \(a_{12} = 8\) (row 1, column 2)
  • \(a_{23} = 9\) (row 2, column 3)

Reading a Matrix from Context

Context What the order tells you
\(3 \times 1\) column matrix 3 items in a single category
\(1 \times 4\) row matrix 4 attributes of a single item
\(n \times n\) square matrix Same number of inputs and outputs

When is Multiplication Defined?

The product \(AB\) is only defined when the number of columns in \(A\) equals the number of rows in \(B\).

\[A: m \times \mathbf{k} \quad B: \mathbf{k} \times n \quad \Rightarrow \quad AB: m \times n\]

Worked Example

Given:

\[C = \begin{pmatrix} 5 & 0 \\ -1 & 3 \\ 2 & 7 \end{pmatrix}\]
  • State the order: \(3 \times 2\)
  • Identify \(c_{31}\): row 3, column 1 \(= 2\)
  • Identify \(c_{22}\): row 2, column 2 \(= 3\)

Can \(CD\) be formed if \(D\) has order \(2 \times 4\)?

Number of columns in \(C = 2\) = number of rows in \(D = 2\). Yes. The product \(CD\) has order \(3 \times 4\).

STUDY HINT: A quick memory aid — “rows come before columns” in the order notation, just as you read left-to-right, top-to-bottom. The subscript \(a_{ij}\) follows the same order: \(i\) = row, \(j\) = column.

VCAA FOCUS: Exam questions often provide two matrices and ask whether their product is defined, and if so, what order the result will be. Check column count of first = row count of second.

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