# Matrix Multiplication: (2×2) by (2×3)

## Introduction

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

A =
 A11 A12 A21 A22

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

B =
 B11 B12 B13 B21 B22 B23

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

A x B =
 A11*B11+A12*B21 A11*B12+A12*B22 A11*B13+A12*B23 A21*B11+A22*B21 A21*B12+A22*B22 A21*B13+A22*B23

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.