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:
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:
Suppose we also have a 3×2 matrix D, which has 3 rows and 2 columns:
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:
Example 2
Suppose we have a 3×3 matrix E, which has 3 rows and 3 columns:
Suppose we also have a 3×2 matrix F, which has 3 rows and 2 columns:
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:
Example 3
Suppose we have a 3×3 matrix G, which has 3 rows and 3 columns:
Suppose we also have a 3×2 matrix H, which has 3 rows and 2 columns:
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:
Additional Resources
The following tutorials explain how to perform other common matrix multiplications:
Matrix Multiplication: (2×2) by (2×2)
Matrix Multiplication: (2×2) by (2×3)