This tutorial shows how to multiply a 3×3 matrix with a 3×2 matrix.
Introduction
Suppose we have a 3×3 matrix A, which has 3 rows and 3 columns:
A =
A11
A12
A13
A21
A22
A23
A31
A32
A33
Suppose we also have a 3×2 matrix B, which has 3 rows and 2 columns:
B =
B11
B12
B21
B22
B31
B32
To multiply matrix A by matrix B, we use the following formula:
A x B =
A11*B11+A12*B21+A13*B31
A11*B12+A12*B22+A13*B32
A21*B11+A22*B21+A23*B31
A21*B12+A22*B22+A23*B32
A31*B11+A32*B21+A33*B31
A31*B12+A32*B22+A33*B32
This results in a 3×2 matrix.
The following examples illustrate how to multiply a 3×3 matrix with a 3×2 matrix using real numbers.
Example 1
Suppose we have a 3×3 matrix C, which has 3 rows and 3 columns:
C =
-3
5
4
1
2
3
-1
0
2
Suppose we also have a 3×2 matrix D, which has 3 rows and 2 columns:
D =
2
1
5
1
0
-1
Here is how to multiply matrix C by matrix D:
C x D =
-3*2 + 5*5 + 4*0
-3*1 + 5*1 + 4*-1
1*2 + 2*5 + 3*0
1*1 + 2*1 + 3*-1
-1*2 + 0*5 + 2*0
-1*1 + 0*1 + 2*-1
This results in the following 3×2 matrix:
C x D =
19
-2
12
0
-2
-3
Example 2
Suppose we have a 3×3 matrix E, which has 3 rows and 3 columns:
E =
2
8
1
3
3
0
0
1
2
Suppose we also have a 3×2 matrix F, which has 3 rows and 2 columns:
F =
-2
-2
3
1
4
10
Here is how to multiply matrix E by matrix F:
E x F =
2*-2 + 8*3 + 1*4
2*-2 + 8*1 + 1*10
3*-2 + 3*3 + 0*4
3*-2 + 3*1 + 0*10
0*-2 + 1*3 + 2*4
0*-2 + 1*1 + 2*10
This results in the following 3×2 matrix:
E x F =
24
14
3
-3
11
21
Example 3
Suppose we have a 3×3 matrix G, which has 3 rows and 3 columns:
G =
-1
0
0
7
1
0
2
4
6
Suppose we also have a 3×2 matrix H, which has 3 rows and 2 columns:
H =
4
5
9
2
0
1
Here is how to multiply matrix G by matrix H:
G x H =
-1*4 + 0*9 + 0*0
-1*5 + 0*2 + 0*1
7*4 + 1*9 + 0*0
7*5 + 1*2 + 0*1
2*4 + 4*9 + 6*0
2*5 + 4*2 + 6*1
This results in the following 3×2 matrix:
G x H =
-4
-5
37
37
44
24
Matrix Calculator
The examples above illustrated how to multiply 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 (3×3) by (3×2)