This tutorial shows how to multiply a 2×2 matrix with a 2×2 matrix.
Introduction
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)