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.
Multiplying Matrices Video Tutorial: (2×2) by (2×3)
