Matrices • Topic 1 of 3

Order of a Matrix

What is the order of a matrix? A matrix is a rectangular arrangement of numbers (called elements) arranged in rows and columns. The order of a matrix describes its size: number of rows × number of columns.

Notation: A matrix with 'm' rows and 'n' columns has order m × n (read as "m by n").

Example: Matrix A = [ 3 5 7 ] has 1 row and 3 columns → order = 1 × 3 Matrix B = [ 2 ] has 1 row and 1 column → order = 1 × 1 Matrix C = [ 1 4 2 5 3 6 ] has 3 rows and 2 columns → order = 3 × 2

General representation: A = [aᵢⱼ] where:

  • i = row number (1 to m)
  • j = column number (1 to n)
  • aᵢⱼ = element in the i-th row and j-th column

Special types of matrices based on order:

TypeDefinitionExample
Row matrixm = 1 (one row)[2, 5, 8]
Column matrixn = 1 (one column)[3; 7; 9]
Square matrixm = n (same rows and columns)[1 2; 3 4]
Zero matrixAll elements are 0[0 0; 0 0]

Real-life analogy: Think of a matrix as a spreadsheet or a table. The order tells you how many rows and columns the table has — just like a classroom seating chart with 5 rows and 4 columns is a 5 × 4 arrangement!

A 3 by 4 matrix with rows and columns labelled, showing order equals number of rows times number of columns Order of a matrix = rows × columns a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 row 1 row 2 row 3 col 1 col 2 col 3 col 4 3 rows 3 rows and 4 columns → order = 3 × 4 (12 entries) Mini grids of the special types of matrices: row, column, square, zero and identity Special types of matrices Row (1×3) Column (3×1) Square (2×2) 0 0 0 0 Zero (O) 1 0 0 1 Identity (I)
Solved Examples
1
Worked Example
Write the order of the following matrices: A = [2 5 8 3], B = [1; 4; 7], C = [1 2; 3 4; 5 6]
Solution
  1. Step 1: Matrix A has 1 row and 4 columns → order = 1 × 4
  2. Step 2: Matrix B has 3 rows and 1 column → order = 3 × 1
  3. Step 3: Matrix C has 3 rows and 2 columns → order = 3 × 2

Answer: A: 1×4, B: 3×1, C: 3×2

2
Worked Example
Construct a 2 × 3 matrix whose elements are given by aᵢⱼ = i + j.
Solution
  1. Step 1: Matrix has 2 rows and 3 columns
  2. Step 2: For i=1, j=1: \(a_{11}\) = 1+1 = 2
  3. Step 3: i=1, j=2: \(a_{12}\) = 1+2 = 3
  4. Step 4: i=1, j=3: \(a_{13}\) = 1+3 = 4
  5. Step 5: i=2, j=1: \(a_{21}\) = 2+1 = 3
  6. Step 6: i=2, j=2: \(a_{22}\) = 2+2 = 4
  7. Step 7: i=2, j=3: \(a_{23}\) = 2+3 = 5

Answer: A = [2 3 4; 3 4 5]

3
Worked Example
If a matrix has 18 elements, what are the possible orders it can have?
Solution
  1. Step 1: Total elements = m × n = 18
  2. Step 2: Find all pairs of positive integers (m, n) such that m × n = 18
  3. Step 3: Factors of 18: 1×18, 2×9, 3×6, 6×3, 9×2, 18×1
  4. Step 4: Each pair represents a possible order (m × n)

Answer: 1×18, 2×9, 3×6, 6×3, 9×2, 18×1

Key Points

  • Order of a matrix = number of rows × number of columns
  • Rows are horizontal, columns are vertical
  • A matrix with m rows and n columns has order m × n
  • Row matrix: m = 1; Column matrix: n = 1
  • Square matrix: m = n
  • Element aᵢⱼ lies in i-th row and j-th column
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The order of a matrix is written as:
Explanation: Rows first, then columns.
Q2.A $2\times3$ matrix has how many elements?
Explanation: $2\times3=6$.
Q3.A matrix with a single row is a:
Explanation: Row matrix.
Q4.The order of $\begin{bmatrix}1&2\\3&4\end{bmatrix}$ is:
Explanation: Two rows, two columns.
Practice more MCQs → 📝 Assignment · 35 marks