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:

A =

A_{11}

A_{12}

A_{13}

A_{21}

A_{22}

A_{23}

A_{31}

A_{32}

A_{33}

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

B =

B_{11}

B_{12}

B_{21}

B_{22}

B_{31}

B_{32}

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

A x B =

A_{11}*B_{11}+A_{12}*B_{21}+A_{13}*B_{31}

A_{11}*B_{12}+A_{12}*B_{22}+A_{13}*B_{32}

A_{21}*B_{11}+A_{22}*B_{21}+A_{23}*B_{31}

A_{21}*B_{12}+A_{22}*B_{22}+A_{23}*B_{32}

A_{31}*B_{11}+A_{32}*B_{21}+A_{33}*B_{31}

A_{31}*B_{12}+A_{32}*B_{22}+A_{33}*B_{32}

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:

C =

-3

5

4

1

2

3

-1

0

2

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

D =

2

1

5

1

0

-1

Here is how to multiply matrix C by matrix D:

C x D =

-3*2 + 5*5 + 4*0

-3*1 + 5*1 + 4*-1

1*2 + 2*5 + 3*0

1*1 + 2*1 + 3*-1

-1*2 + 0*5 + 2*0

-1*1 + 0*1 + 2*-1

This results in the following 3×2 matrix:

C x D =

19

-2

12

0

-2

-3

Example 2

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

E =

2

8

1

3

3

0

0

1

2

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

F =

-2

-2

3

1

4

10

Here is how to multiply matrix E by matrix F:

E x F =

2*-2 + 8*3 + 1*4

2*-2 + 8*1 + 1*10

3*-2 + 3*3 + 0*4

3*-2 + 3*1 + 0*10

0*-2 + 1*3 + 2*4

0*-2 + 1*1 + 2*10

This results in the following 3×2 matrix:

E x F =

24

14

3

-3

11

21

Example 3

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

G =

-1

0

0

7

1

0

2

4

6

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

H =

4

5

9

2

0

1

Here is how to multiply matrix G by matrix H:

G x H =

-1*4 + 0*9 + 0*0

-1*5 + 0*2 + 0*1

7*4 + 1*9 + 0*0

7*5 + 1*2 + 0*1

2*4 + 4*9 + 6*0

2*5 + 4*2 + 6*1

This results in the following 3×2 matrix:

G x H =

-4

-5

37

37

44

24

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)