Matrix Multiplication: (3×3) by (3×2)

Matrix multiplication tutorial

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)

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