Matrix Multiplication: (3×3) by (3×2)

This tutorial shows how to multiply a 3×3 matrix with a 3×2 matrix.

Introduction

Suppose we have a 3×3 matrix A, which has 3 rows and 3 columns:

Suppose we also have a 3×2 matrix B, which has 3 rows and 2 columns:

To multiply matrix A by matrix B, we use the following formula:

This results in a 3×2 matrix.

The following examples illustrate how to multiply a 3×3 matrix with a 3×2 matrix using real numbers.

Example 1

Suppose we have a 3×3 matrix C, which has 3 rows and 3 columns:

Suppose we also have a 3×2 matrix D, which has 3 rows and 2 columns:

Here is how to multiply matrix C by matrix D:

This results in the following 3×2 matrix:

Example 2

Suppose we have a 3×3 matrix E, which has 3 rows and 3 columns:

Suppose we also have a 3×2 matrix F, which has 3 rows and 2 columns:

Here is how to multiply matrix E by matrix F:

This results in the following 3×2 matrix:

Example 3

Suppose we have a 3×3 matrix G, which has 3 rows and 3 columns:

Suppose we also have a 3×2 matrix H, which has 3 rows and 2 columns:

Here is how to multiply matrix G by matrix H:

This results in the following 3×2 matrix:

Matrix Calculator

The examples above illustrated how to multiply matrices by hand. A good way to double check your work if you’re multiplying matrices by hand is to confirm your answers with a matrix calculator. While there are many matrix calculators online, the simplest one to use that I have come across is this one by Math is Fun.

Multiplying Matrices Video Tutorial (3×3) by (3×2)