Matrices: Definition

Key idea

A rectangular array of numbers with m rows and n columns.

In depth

A matrix is a rectangular array of numbers with m rows and n columns (m × n). It stores data, systems of equations, or a linear transformation. Entry aᵢⱼ is in row i, column j.

Worked example

[1 2 3; 4 5 6] is a 2 × 3 matrix; a₂₃ = 6.

Formula

Real-world uses

• Storing tables of data such as exam marks by student and subject in spreadsheets and code

Common mistake

Mixing up the row and column indices.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related intermediate topics in Linear Algebra

• Scalars and Vectors – A scalar is a number; a vector has magnitude and direction.
• Vector Notation and Components – Write vectors as ordered lists of coordinates.
• Vector Addition – Add component-wise; geometrically tip-to-tail.
• Scalar Multiplication – Scale a vector’s length.
• Vector Magnitude – Length of a vector.
• Unit Vectors – Vectors of length 1, found by normalising.