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 2×2 matrix B, which has 2 rows and 2 columns:

B =

B_{11}

B_{12}

B_{21}

B_{22}

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

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

This results in a 2×2 matrix.

The following examples illustrate how to multiply a 2×2 matrix with a 2×2 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 2×2 matrix D, which has 2 rows and 2 columns:

D =

2

1

5

1

Here is how to multiply matrix C by matrix D:

C x D =

7*2 + 5*5

7*1 + 5*1

6*2 + 3*5

6*1 + 3*1

This results in the following 2×2 matrix:

C x D =

39

12

27

9

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 2×2 matrix F, which has 2 rows and 2 columns:

F =

3

6

2

4

Here is how to multiply matrix E by matrix F:

E x F =

-2*3 + 4*2

-2*6 + 4*4

9*3 + 2*2

9*6 + 2*4

This results in the following 2×2 matrix:

E x F =

2

4

31

62

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 2×2 matrix H, which has 2 rows and 2 columns:

H =

1

2

4

5

Here is how to multiply matrix G by matrix H:

G x H =

2*1 + 3*4

2*2 + 3*5

4*1 + 5*4

4*2 + 5*5

This results in the following 2×2 matrix:

G x H =

14

19

24

33

Example 4

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

I =

1

3

5

7

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

J =

2

4

6

8

Here is how to multiply matrix I by matrix J:

I x J =

1*2 + 3*6

1*4 + 3*8

5*2 + 7*6

5*4 + 7*8

This results in the following 2×2 matrix:

I x J =

20

28

52

76

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 (2×2)