Multiplying Matrices

Summary

This video tutorial delves into the concept of matrix multiplication, emphasizing the importance of order and demonstrating how it affects the resulting matrix's dimensions. It illustrates the process using two examples, first with 1x3 and 3x1 matrices to produce a 1x1 matrix, and then with 2x3 and 3x4 matrices to yield a 2x4 matrix. The video clarifies that matrix multiplication is not commutative, meaning AB ≠ BA, and guides viewers through the step-by-step calculation, reinforcing the rules with practical examples.

Takeaways

• 📚 Matrix multiplication is a process where you multiply the elements of rows from the first matrix by the elements of columns from the second matrix and sum the products.
• 🔄 The order of matrix multiplication matters; AB and BA are not necessarily the same due to the requirement of matching the number of columns in the first matrix with the number of rows in the second.
• 🔢 The order of a matrix is expressed as rows by columns, and it's essential for determining if two matrices can be multiplied and what the resulting matrix's order will be.
• 📏 For matrix A (1x3) and matrix B (3x1), the product AB is a 1x1 matrix, while BA is not possible because the number of columns in A does not match the number of rows in B.
• 🤔 When multiplying matrices, ensure the number of columns in the first matrix equals the number of rows in the second matrix to perform the operation.
• 📝 The resulting matrix's order from the multiplication of two matrices is the product of the first matrix's rows and the second matrix's columns.
• 🧩 In the example given, multiplying matrix A (1x3) by matrix B (3x1) results in a single element (-4), demonstrating a 1x1 matrix.
• 🔍 For matrix B (3x1) and matrix A (1x3), the multiplication BA results in a 3x3 matrix, showing how each element is the sum of products of corresponding row and column elements.
• 📉 The script provides a step-by-step guide to multiplying matrices, illustrating the process with specific numerical examples.
• 📈 The example with matrix A (2x3) and matrix B (3x4) shows that AB is a 2x4 matrix, while BA is not possible due to the mismatch in the number of columns and rows.
• 📝 The multiplication of matrix A (2x3) by matrix B (3x4) is detailed, emphasizing the process of calculating each entry in the resulting 2x4 matrix.

Q & A

What is the main focus of the video?

The main focus of the video is on multiplying matrices and understanding the importance of the order in which matrices are multiplied.

What are the elements of Matrix A and Matrix B in the first example?

In the first example, Matrix A has elements 2, 5, and 6, and Matrix B has elements 3, 4, and -5.

Why is the order of multiplication important in matrix multiplication?

The order of multiplication is important because it affects the resulting matrix's dimensions and values. AB and BA can yield different results and may even have different dimensions if they are not square matrices.

What is the order of Matrix A in the first example?

The order of Matrix A in the first example is 1x3, meaning it has 1 row and 3 columns.

What is the order of Matrix B in the first example?

The order of Matrix B in the first example is 3x1, meaning it has 3 rows and 1 column.

What is the resulting order of the product of Matrix A and Matrix B in the first example?

The resulting order of the product of Matrix A and Matrix B in the first example is 1x1, as the number of columns in Matrix A equals the number of rows in Matrix B.

What is the result of multiplying Matrix A by Matrix B in the first example?

The result of multiplying Matrix A by Matrix B in the first example is a 1x1 matrix with the single value of -4.

Why can't we multiply Matrix B by Matrix A in the first example?

We can't multiply Matrix B by Matrix A in the first example because the number of columns in Matrix B (1) does not equal the number of rows in Matrix A (3), which is a requirement for matrix multiplication.

What is the resulting order of the product of a 2x3 matrix and a 3x4 matrix?

The resulting order of the product of a 2x3 matrix and a 3x4 matrix is a 2x4 matrix, as the number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be possible.

Why is it not possible to multiply a 3x4 matrix by a 2x3 matrix?

It is not possible to multiply a 3x4 matrix by a 2x3 matrix because the number of columns in the 3x4 matrix (4) does not match the number of rows in the 2x3 matrix (2), which violates the requirement for matrix multiplication.

How does the video demonstrate the process of matrix multiplication?

The video demonstrates the process of matrix multiplication by taking elements from the corresponding row of the first matrix and multiplying them with the elements of the corresponding column of the second matrix, then summing these products to get the entry in the resulting matrix.

Outlines

00:00

🧠 Matrix Multiplication Basics

This paragraph introduces the concept of matrix multiplication, focusing on the importance of order when multiplying matrices. It provides an example with two matrices, A and B, and demonstrates that the product AB is not the same as BA. The paragraph explains the order of matrices and how it affects the resulting product, emphasizing that the number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be possible. The example concludes with the multiplication of a 1x3 matrix A by a 3x1 matrix B, resulting in a 1x1 matrix.

05:03

🔍 Exploring Matrix Order and Multiplication

The second paragraph delves deeper into matrix multiplication, illustrating the process with a 1x3 matrix A and a 3x1 matrix B to show that the resulting matrix AB is a 1x1 matrix, while BA would be a 3x3 matrix if it were possible. The paragraph explains the rules governing matrix order and multiplication, emphasizing that the number of columns in the first matrix must equal the number of rows in the second matrix. It also provides a new example with two matrices, A and B, and asks the viewer to determine the order of the products AB and BA, highlighting that BA is not possible due to the mismatch in the number of columns and rows.

10:04

📚 Step-by-Step Matrix Multiplication

This paragraph provides a detailed step-by-step guide on how to multiply two matrices, A and B. It explains that the order of the resulting matrix is determined by the number of rows in the first matrix and the number of columns in the second matrix. The example given involves a 2x3 matrix A and a 3x4 matrix B, which can be multiplied to form a 2x4 matrix. The paragraph walks through the multiplication process, showing how to calculate each entry of the resulting matrix by taking the dot product of rows from the first matrix and columns from the second.

15:08

📝 Completing Matrix Multiplication with Examples

The final paragraph continues the explanation of matrix multiplication, completing the example from the previous paragraph. It shows the calculation for each entry of the resulting 2x4 matrix by multiplying corresponding elements from the rows of matrix A and the columns of matrix B. The paragraph emphasizes the importance of following the correct order of multiplication and placing the results in the correct position in the new matrix. It concludes by summarizing the process and confirming that the resulting matrix is indeed 2x4, as predicted by the initial analysis of the matrices' orders.

Keywords

💡Matrix

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of the video, matrices are the primary objects for multiplication, and understanding their structure is fundamental to grasping the video's theme of matrix multiplication.

💡Matrix Multiplication

Matrix multiplication is an operation that takes a pair of matrices and produces a new matrix by combining the values of the input matrices in a specific way. The video focuses on this process, demonstrating how to calculate the product of two matrices, A and B, and emphasizing the importance of the order of multiplication.

💡Order of Matrix

The order of a matrix refers to the number of rows and columns it contains, typically expressed as 'm x n', where 'm' is the number of rows and 'n' is the number of columns. The video script explains how the order of the resulting matrix in a multiplication operation is determined by the orders of the matrices being multiplied.

💡Element

In the context of matrices, an element is a single number or value that occupies a space within the matrix at a specific row and column intersection. The script uses elements of matrices A and B to illustrate the process of matrix multiplication.

💡Row

A row in a matrix is a horizontal sequence of elements. The video script describes how to multiply elements of a row by the corresponding elements of a column during matrix multiplication, which is essential for understanding the calculation process.

💡Column

A column in a matrix is a vertical sequence of elements. The script explains that during matrix multiplication, you multiply the elements of a row by the corresponding elements of a column, emphasizing the role of columns in determining the structure of the resulting matrix.

💡Diagonal

The diagonal of a matrix refers to the elements that run from the top left to the bottom right. In the video, the diagonal is implicitly mentioned when discussing the multiplication of corresponding row and column elements, which can result in a diagonal entry in the product matrix.

💡Scalar

A scalar is a single number that can be the result of an operation involving matrices, such as the product of a row and a column in matrix multiplication. The script mentions scalar results when calculating the entries of the resulting matrix from the product of matrices A and B.

💡Identity Matrix

An identity matrix is a special square matrix in which all the elements of the main diagonal are ones and all other elements are zeros. While not explicitly mentioned in the script, the concept is relevant as the multiplication of a matrix by the identity matrix results in the original matrix, which could be a point of comparison for understanding matrix multiplication.

💡Transpose

The transpose of a matrix is an operation that flips the matrix over its diagonal, swapping its rows and columns. Although not directly discussed in the script, the concept of transposing is relevant to understanding how matrix multiplication affects the order of elements and the resulting matrix's dimensions.

💡Determinant

The determinant is a scalar value that can be computed from a square matrix and has important properties in linear algebra. While not discussed in the script, the determinant is related to the invertibility of matrices, which is a concept that could be connected to matrix multiplication and the existence of a product.

Highlights

The video focuses on matrix multiplication and the importance of order in matrix operations.

Matrix A is defined with elements 2, 5, and 6, and Matrix B with elements 3, 4, and -5.

Multiplication of 3 by 5 and 5 by 3 is the same in arithmetic but not in matrix multiplication due to order sensitivity.

Matrix A is a 1x3 matrix, and Matrix B is a 3x1 matrix, illustrating the concept of matrix dimensions.

The product of A and B (AB) will be a 1x1 matrix, demonstrating matrix multiplication rules.

The order of AB is determined by the number of rows in A and columns in B, resulting in a 1x1 matrix.

The order of BA is 3x3, showing that matrix dimensions change based on multiplication order.

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second.

The process of multiplying matrix A by matrix B is demonstrated step by step.

The result of AB is a single element, -4, showing the outcome of a 1x1 matrix multiplication.

The multiplication of BA is shown to result in a 3x3 matrix, contrasting with AB.

The video provides a clear example of how matrix dimensions are calculated and the resulting matrix's order.

A second example with matrices A and B of different dimensions is introduced for further illustration.

Matrix A is a 2x3 matrix, and Matrix B is a 3x4 matrix, setting the stage for another multiplication example.

The multiplication of A by B is possible due to matching dimensions, but B by A is not, highlighting dimension requirements.

The resulting matrix AB is a 2x4 matrix, as predicted by the rules of matrix multiplication.

A detailed step-by-step multiplication of A by B is provided, including intermediate calculations.

The final result of the multiplication is a 2x4 matrix, confirming the theoretical predictions.

Transcripts

in this video we're going to focus on

multiplying matrices

so let's say if we have matrix a

and it has the elements

2

5 and 6

and matrix b

contains the elements 3 4

and negative 5.

so let's find the product of a b

and b a

now in math

let's say if we multiply 3 times 5

is 15

and 5 times 3 is also 15.

when dealing with matrices the order

matters

a b and b a

they will not be the same

so keep that in mind

a b tells us that we need to take matrix

a

and then multiply by b in that order

b a tells us that we need to start with

matrix b and then multiply by a

so the order matters

before we begin multiplying let's talk

about the order of each matrix what is

the order of matrix a

matrix a

has one row

and

three columns

so it's a one by three matrix

the way you write the order is you start

with the rows

and then

you multiply by the columns

so matrix b

has three rows

and only one column

so it's a three by one

matrix

so if we multiply

a by b

what do you think the order will be

so a times b

so a is a one by three matrix

and b is a three by one matrix

in order to multiply these two matrices

the columns

in the first matrix has to equal the

number of rows in the second matrix

so make sure you understand that

and the order

of matrix a b is going to be the product

of these two numbers which

do not have to be the same

so the order of a b

is going to be a one by one matrix

so therefore we should have one number

in this matrix only it could be like 7

12 15 who knows

but the order of the matrix

is basically the product of those two

now what about b times a

what is the order of that matrix so we

take matrix b

and multiply by matrix a what should we

get

so matrix b is a three by one matrix

and matrix a

is

a one by three matrix

so we have to make sure that

the number of columns in the first

matrix

is equal

to the number of rows in the second

matrix

so which we do have so we can multiply

these two matrices

now the order of the matrix is going to

be 3 by 3.

so the order for matrix b a it's going

to be a three by three matrix

and for a b we said it's going to be a

one by one matrix

now let's confirm that

so let's get rid of

some stuff

so let's start by multiplying

matrix a by matrix b so the way you do

it is you take all the elements in the

first row and then multiply it by all

the elements

in the first column

so it's always going to be row times the

column

and then you add the products together

so it's always going to be r times c

so we're going to take

this one it's in the first row first

column and multiply it by

that element in the first row and the

first column so it's going to be 2 times

3

and then we're going to multiply these

two

so it's going to be 5 times 4

and then

six times negative five and this will

give us only one element

so because we took the elements in the

first row

and multiplied it by the elements in the

first column

we're going to get a single entry

in the first row first column

so it's going to be a one by one matrix

now two times three

is six

five times four is twenty

six times negative five that's negative

thirty

and six plus twenty is twenty six

twenty six minus thirty

is negative four

so we can see that

a b is a one by one matrix it has

one row and only one column

and so it's equal to negative four

now let's multiply b by a

so let's show that it's going to be a

three by three matrix

so we're going to start with

b so remember it's row by column

so the first row

times the first column so that's going

to be three times two

and that entry

is going to go in the first row first

column

so three times two is six

so now we're still on row one but now we

have to multiply by column two

so that's going to be three times five

so it's going to be in the first row but

second column so that's going to be 15.

and then we need to multiply row one

by column three

so so it's going to be row by column

so three times six is 18.

next we're gonna move on to row two and

then column one

four times two is eight

so that's going to go in the second row

first column

and then

we're gonna have

row two by column two

so that's four times five

and that's twenty

and then it's going to be

row two times column three

and four times six

is

twenty-four

so hopefully you're seeing the pattern

of what we're doing here

if not feel free to pause the video and

rewind until you understand it and then

it's going to be row 3 times column 1.

so negative 5 times 2 is negative 10

and then row 3 times column 2

5 times negative 5

is

negative 25

and then row 3

times column 3.

6 times negative 5 is

negative 30.

so as we could see

a b and b a are clearly different

and you can see that ba

has three rows

and

three columns so it's a three by three

matrix

so hopefully this example gave you a

good idea of how to multiply matrices

let's work on another example

so let's say that

matrix a

contains the elements 1

4 negative 2

3 5 and negative 6

and matrix b

contains the elements

5 2

8 negative 1

3 6 4

5

negative 2

9

7 and negative 3.

so go ahead

and

multiply

a times b and also b times a

and determine the order

of

these two products

so what is the order of a b

so what do you think the order is going

to be

so notice that a has two rows

and it has

three columns

so matrix a

is

it's a two by three matrix it has two

rows and three columns

matrix b

contains

three rows

and it has

four columns

so matrix b is a

three by four matrix

so notice that these numbers are the

same

the number of columns

in

matrix a is equal to the number of rows

in matrix b

so that means that we are allowed to

multiply a by b

these two numbers

they must be the same

in order

for you to multiply the two matrices

now the order of matrix a b

is going to be 2 times 4 or 2 by 4.

so it's going to have two rows and four

columns

now what about b times a

what can we say about that

so matrix b

has

three rows and four columns

and matrix a

has two rows and three columns so notice

that these two numbers

do not match

they're different

and in order to multiply b times a they

must be the same so therefore

b a does not exist

we don't have to worry about that

problem

so let's focus on multiplying a by b

so it's going to have two rows

and four columns

so let's start by multiplying

row one

by column one

so it's going to be 1 times 5

and then it's going to be plus

4 times 3

and then

plus negative 2 times negative 2.

so you got to multiply these two first

and then these two together

and then those two

it might be wise for me to perform the

operation somewhere else

so let me get rid of this

i'm going to put the answers there so

it's going to be

1 times 5

plus 4 times

3

plus negative two times negative two

so this is gonna be five plus twelve

plus four

five plus twelve is seventeen seventeen

plus four is twenty one

so because we multiplied row one

by column one

we need to put this answer

in

row one column one

so this is row one row two

and this is going to be

row column one two three and four

now let's multiply row one

by column two

and that will produce an entry in

row one column two

so first we need to take

one

and then multiply it by two

and then it's going to be 4

times 6

and then negative 2

times

9. so 1 times 2 is 2 4 times 6 is 24

negative 2 times 9 is negative 18.

so 2 plus 24 is 26

26 minus 18 is 8.

so let's place that entry

in that slot

now let's move on to

row one

column three

and that's going to be an entry in row

one column three

so we're going to multiply one

by eight

and then four

by four

and then negative two times seven

or you can write it as

plus

negative two times seven

so one times eight is eight four times

four is sixteen

negative two times seven

is negative fourteen

now sixteen minus fourteen is two

a plus two is ten

so we get that answer

now let's move on to the first row

fourth column

which will produce an entry in row one

column four

so it's going to be one

times negative one

plus

four times five

plus negative two times negative three

so this is going to be negative one four

times five is twenty

negative two times negative three is

plus six

so twenty plus six is twenty six

and twenty six minus one is twenty

so hopefully by now

you get the hang of it if not feel free

to rewind the video

so let's move a little faster

so row two times

column one

that's going to be 3

times 5

plus

5 times 3

plus negative 6 times negative 2.

so that's 15 plus 15

plus 12. 15 and 15 is 30

30 plus 12 is 42

so we need to put that there

next we have

row 2

times column two

so that's going to be three times two

plus five times six

plus negative six

times nine three and two is six five

times six is thirty

negative six times nine is negative

fifty four

now thirty minus fifty four is negative

twenty four

and six minus twenty four

is

negative eighteen

now let's move on to row two

times column three

so it's going to be three times eight

plus five times four

plus negative six times seven

so three times eight is twenty-four five

times four is twenty

negative six times seven is

negative forty-two

twenty-four plus twenty is 44

and 44 minus 42 is 2.

now for the last entry

we're going to multiply

row 2

by column 4.

so that's going to be three

times negative one

plus five times five

plus negative six times negative three

so this is negative three

five times five is 25 and negative six

times negative three is eighteen

now negative three plus twenty-five is

twenty-two

twenty-two plus eighteen is forty

and so now you know how to multiply

two matrices together

and as you can see

we did get the two by four matrix as

predicted

so if you multiply

a two by three matrix

with a three by four matrix

first you must see that

these two are the same

and then the order of the new matrix is

going to be

the number of rows of the first matrix

times the columns of the second which we

have here it's two rows and four columns

you