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)