Suppose we have a 2×2 matrix A, which has 2 rows and 2 columns:

A =

A_{11}

A_{12}

A_{21}

A_{22}

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

B =

B_{11}

B_{12}

B_{13}

B_{21}

B_{22}

B_{23}

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

A x B =

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

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

A_{11}*B_{13}+A_{12}*B_{23}

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

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

A_{21}*B_{13}+A_{22}*B_{23}

This results in a 2×3 matrix.

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

Example 1

Suppose we have a 2×2 matrix C, which has 2 rows and 2 columns:

C =

7

5

6

3

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

D =

2

1

4

5

1

2

Here is how to multiply matrix C by matrix D:

C x D =

7*2 + 5*5

7*1 + 5*1

7*4 + 5*2

6*2 + 3*5

6*1 + 3*1

6*4 + 3*2

This results in the following matrix:

C x D =

39

12

38

27

9

30

Example 2

Suppose we have a 2×2 matrix E, which has 2 rows and 2 columns:

E =

-2

4

9

2

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

F =

3

6

9

2

4

6

Here is how to multiply matrix E by matrix F:

E x F =

-2*3 + 4*2

-2*6 + 4*4

-2*9 + 4*6

9*3 + 2*2

9*6 + 2*4

9*9 + 2*6

This results in the following matrix:

E x F =

2

4

6

31

62

93

Example 3

Suppose we have a 2×2 matrix G, which has 2 rows and 2 columns:

G =

2

3

4

5

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

H =

1

2

3

4

5

6

Here is how to multiply matrix G by matrix H:

G x H =

2*1 + 3*4

2*2 + 3*5

2*3 + 3*6

4*1 + 5*4

4*2 + 5*5

4*3 + 5*6

This results in the following matrix:

G x H =

14

19

24

24

33

42

Matrix Calculator

The examples above illustrated how to multiply 2×2 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: (2×2) by (2×3)

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.