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 = 

A_{11} 
A_{12} 
A_{13} 

A_{21} 
A_{22} 
A_{23} 
A_{31} 
A_{32} 
A_{33} 

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

B_{11} 
B_{12} 

B_{21} 
B_{22} 
B_{31} 
B_{32} 

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

A_{11}*B_{11}+A_{12}*B_{21}+A_{13}*B_{31} 
A_{11}*B_{12}+A_{12}*B_{22}+A_{13}*B_{32} 

A_{21}*B_{11}+A_{22}*B_{21}+A_{23}*B_{31} 
A_{21}*B_{12}+A_{22}*B_{22}+A_{23}*B_{32} 
A_{31}*B_{11}+A_{32}*B_{21}+A_{33}*B_{31} 
A_{31}*B_{12}+A_{32}*B_{22}+A_{33}*B_{32} 

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)