What is the result of multiplying a (2×2) matrix by a (2×3) matrix?

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The result of multiplying a (2×2) matrix by a (2×3) matrix is a (2×3) matrix. This means that the resulting matrix will have two rows and three columns, with each element being the product of the corresponding elements from the two original matrices. This operation is known as matrix multiplication and is commonly used in various mathematical and computational applications.

Matrix Multiplication: (2×2) by (2×3)

This tutorial shows how to multiply a 2×2 matrix with a 2×3 matrix.

Introduction

Suppose we have a 2×2 matrix A, which has 2 rows and 2 columns:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.super_short2 {
max-width: 180px;
margin: 5px auto;
color: blue;
}

A =
  A11 A12  
A21 A22

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

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.short2 {
max-width: 250px;
margin: 5px auto;
color: red;
}

B =
  B11 B12 B13  
B21 B22 B23

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

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.long{
margin: 5px auto;
color: #000000;
}
.red {
color: red;
}
.blue {
color: blue;
}

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:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.super_short {
max-width: 150px;
margin: 5px auto;
color: #000000;
}

C =
  7 5  
6 3

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

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.short {
max-width: 250px;
margin: 5px auto;
color: #000000;
}

D =
  2 1 4  
5 1 2

Here is how to multiply matrix C by matrix D:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.medium {
max-width: 500px;
margin: 5px auto;
color: #000000;
}
.red {
color: red;
}
.blue {
color: blue;
}

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:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.short {
max-width: 250px;
margin: 5px auto;
color: #000000;
}

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:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.super_short {
max-width: 150px;
margin: 5px auto;
color: #000000;
}

E =
  -2 4  
9 2

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

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.short {
max-width: 250px;
margin: 5px auto;
color: #000000;
}

F =
  3 6 9  
2 4 6

Here is how to multiply matrix E by matrix F:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.medium {
max-width: 500px;
margin: 5px auto;
color: #000000;
}
.red {
color: red;
}
.blue {
color: blue;
}

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:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.short {
max-width: 250px;
margin: 5px auto;
color: #000000;
}

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:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.super_short {
max-width: 150px;
margin: 5px auto;
color: #000000;
}

G =
  2 3  
4 5

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

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.short {
max-width: 250px;
margin: 5px auto;
color: #000000;
}

H =
  1 2 3  
4 5 6

Here is how to multiply matrix G by matrix H:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.medium {
max-width: 500px;
margin: 5px auto;
color: #000000;
}
.red {
color: red;
}
.blue {
color: blue;
}

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:

table {
border-collapse: collapse;
border-spacing: 0;
padding: 0;
}
td.tdleft {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-left: solid 1px #000;
width: 5px;
padding: 0;
}
td.tdreg {
padding: 2px 1px;
text-align: center;
border-bottom: solid 1px #fff;
}
td.tdright {
border-top: solid 1px #000;
border-bottom: solid 1px #000;
border-right: solid 1px #000;
width: 5px;
padding: 0;
}
.short {
max-width: 250px;
margin: 5px auto;
color: #000000;
}

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)

https://www.youtube.com/watch?v=yeKJbi8-heE
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