Matrix Definition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Matrix Definition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A matrix is a rectangular array of numbers arranged in rows (horizontal) and columns (vertical). An m \times n matrix has m rows and n columns. Each number in the matrix is called an entry or element, identified by its row and column position.

Think of a spreadsheet: rows go across, columns go down, and every cell holds a number. A 2 \times 3 matrix is like a mini-spreadsheet with 2 rows and 3 columns. Matrices package multiple numbers into a single organized object so you can manipulate them all at once.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Matrices organize data into a grid structure that enables powerful operations on entire systems of numbers simultaneously.

Common stuck point: Dimensions are always rows \times columns, never the reverse. A 3 \times 2 matrix has 3 rows and 2 columns, not 2 rows and 3 columns.

Sense of Study hint: Count rows first (horizontal lines), then columns (vertical lines). Write the dimensions as rows x columns.

Worked Examples

Example 1

easy
Given A = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix}, what are the dimensions of A and what is a_{2,3}?

Solution

  1. 1
    Step 1: Count rows: 2. Count columns: 3. Dimensions: 2 \times 3.
  2. 2
    Step 2: a_{2,3} means row 2, column 3. That entry is 2.
  3. 3
    Step 3: Verify: row 2 is [0, 5, 2], third element is 2 ✓

Answer

A is 2 \times 3; a_{2,3} = 2
A matrix's dimensions are always given as rows × columns. The notation a_{i,j} refers to the entry in row i and column j.

Example 2

medium
Write a 3 \times 1 column matrix where a_{i,1} = 2i - 1.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
What are the dimensions of \begin{bmatrix} 4 & 0 \\ -1 & 7 \\ 3 & 2 \end{bmatrix}?

Example 2

medium
If A is a 4 \times 3 matrix, how many entries does it have? Can you multiply A by a 3 \times 5 matrix?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

systems of equationsvariables