Matrix Multiplication: (2×2) by (2×2)

Matrix multiplication tutorial

This tutorial shows how to multiply a 2×2 matrix with a 2×2 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 2×2 matrix B, which has 2 rows and 2 columns:

B =
B11 B12
B21 B22

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

A x B =
A11*B11+A12*B21 A11*B12+A12*B22
A21*B11+A22*B21 A21*B12+A22*B22

This results in a 2×2 matrix.

The following examples illustrate how to multiply a 2×2 matrix with a 2×2 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 2×2 matrix D, which has 2 rows and 2 columns:

D =
2 1
5 1

Here is how to multiply matrix C by matrix D:

C x D =
7*2 + 5*5 7*1 + 5*1
6*2 + 3*5 6*1 + 3*1

This results in the following 2×2 matrix:

C x D =
39 12
27 9

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 2×2 matrix F, which has 2 rows and 2 columns:

F =
3 6
2 4

Here is how to multiply matrix E by matrix F:

E x F =
-2*3 + 4*2 -2*6 + 4*4
9*3 + 2*2 9*6 + 2*4

This results in the following 2×2 matrix:

E x F =
2 4
31 62

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 2×2 matrix H, which has 2 rows and 2 columns:

H =
1 2
4 5

Here is how to multiply matrix G by matrix H:

G x H =
2*1 + 3*4 2*2 + 3*5
4*1 + 5*4 4*2 + 5*5

This results in the following 2×2 matrix:

G x H =
14 19
24 33

Example 4

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

I =
1 3
5 7

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

J =
2 4
6 8

Here is how to multiply matrix I by matrix J:

I x J =
1*2 + 3*6 1*4 + 3*8
5*2 + 7*6 5*4 + 7*8

This results in the following 2×2 matrix:

I x J =
20 28
52 76

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 (2×2)

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