Matrix Multiplication: (2×2) by (2×3)

Matrix multiplication tutorial

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:

A =
A11 A12
A21 A22

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

B =
B11 B12 B13
B21 B22 B23

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

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:

C =
7 5
6 3

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

D =
2 1 4
5 1 2

Here is how to multiply matrix C by matrix D:

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:

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:

E =
-2 4
9 2

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

F =
3 6 9
2 4 6

Here is how to multiply matrix E by matrix F:

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:

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:

G =
2 3
4 5

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

H =
1 2 3
4 5 6

Here is how to multiply matrix G by matrix H:

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:

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)

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