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

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

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

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 = 

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

Here is how to multiply matrix C by matrix D:
C x D = 

This results in the following matrix:
C x D = 

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

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

Here is how to multiply matrix E by matrix F:
E x F = 

This results in the following matrix:
E x F = 

Example 3
Suppose we have a 2×2 matrix G, which has 2 rows and 2 columns:
G = 

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

Here is how to multiply matrix G by matrix H:
G x H = 

This results in the following matrix:
G x H = 

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.