Intro to Matrices

Summary

This video tutorial introduces the concept of matrices as arrays of numbers arranged in rows and columns. It explains how to determine a matrix's order, identify specific elements, and distinguish square matrices. The script also covers basic matrix operations, including addition and multiplication by a scalar, as well as subtraction, providing examples to illustrate the processes clearly.

Takeaways

• 📊 A matrix is an array of numbers organized into rows and columns.
• 🔢 The order of a matrix is defined by the number of rows followed by the number of columns, e.g., a 2x3 matrix has two rows and three columns.
• 📍 Elements in a matrix are identified by their row and column, like 'a23' for the element in the second row and third column.
• 🔑 To find the value of a specific matrix element, identify its position by row and column numbers.
• 🧩 Matrix order must match when performing addition or subtraction of matrices; they must have the same dimensions.
• ➕ Adding matrices involves summing corresponding elements from each matrix.
• 🔍 Multiplying a matrix by a scalar (e.g., 4 times matrix A) means multiplying every element of the matrix by that scalar.
• ➖ Subtracting one matrix from another (A - B) is done by subtracting the corresponding elements of matrix B from A.
• 🔄 The process of identifying matrix elements and performing operations like addition, subtraction, and scalar multiplication is fundamental in matrix algebra.
• 🔍 Square matrices have an equal number of rows and columns, making them distinct from other matrix shapes.
• 📚 Understanding matrix operations is crucial for further studies in pre-calculus and linear algebra.

Q & A

What is a matrix?

A matrix is an array of numbers organized into rows and columns.

How is the order of a matrix defined?

The order of a matrix is defined by the number of rows followed by the number of columns it contains, such as a 'two by three' matrix for a 2x3 matrix.

What does the term 'element' in a matrix refer to?

An 'element' in a matrix refers to a specific number within the matrix, identified by its row and column indices, like 'a23' for the element in the second row and third column.

How do you identify the value of element a12 in a matrix?

To identify the value of element a12, locate the number in the first row and second column of the matrix.

What is a square matrix?

A square matrix is a matrix where the number of rows is equal to the number of columns, such as a 2x2 or 3x3 matrix.

How do you determine the order of matrix B with the given elements?

Matrix B with the given elements has three rows and four columns, making it a 'three by four' matrix.

What is the value of element b23 in the described matrix B?

The value of element b23 in matrix B is -4, as it is located in the second row and third column.

How do you add two matrices together?

To add two matrices, add the corresponding elements of the same row and column. The matrices must have the same order to be added together.

What is the result of multiplying every element in matrix A by 4?

Multiplying every element in matrix A by 4 results in a new matrix with elements that are four times the original elements, such as 8, 12, 20, and -16 for the given matrix A.

How do you subtract one matrix from another?

To subtract one matrix from another, subtract the corresponding elements of the same row and column. The matrices must have the same order to be subtracted.

What is the difference between matrix A and B if matrix A is [2, 3, 5, -4] and matrix B is [7, 4, -3, 5]?

The difference between matrix A and B is a new matrix with elements calculated as A - B, which results in [-5, -1, 8, -9].

Outlines

00:00

📊 Understanding Matrices and Their Elements

This paragraph introduces the concept of matrices as arrays of numbers arranged in rows and columns. It explains the order of a matrix by using 'matrix A' with the example of a 2x3 matrix, detailing how to identify its specific elements like a23 and a12. The paragraph also covers identifying the order of 'matrix B', a 3x4 matrix, and finding the values of its elements such as b11, b23, b14, and b34. The task for the viewer is to identify the order of additional matrices C, D, E, F, and G, and to recognize square matrices among them.

05:01

🔢 Determining Matrix Orders and Performing Basic Operations

This section continues the discussion on matrices by identifying the orders of matrices C through H, which are 2x2, 3x2, 1x1, 1x4, and 3x3 respectively. It emphasizes that a square matrix has an equal number of rows and columns, like matrix C and G. The paragraph then explains how to add two matrices of the same order, using matrices A and B as examples, by adding corresponding elements to get the sum. It also demonstrates matrix multiplication by a scalar, specifically multiplying every element of matrix A by four, resulting in 4A with new values.

10:02

➖ Subtracting Matrices and Additional Resources

The final paragraph covers the process of subtracting one matrix from another, using matrices A and B to illustrate. It shows the step-by-step subtraction of corresponding elements to find the difference, resulting in a new matrix with values such as -5, -1, 8, and -9. The paragraph concludes by directing viewers to the description section for more pre-calculus videos and expresses gratitude for watching the video.

Keywords

💡Matrix

A matrix is a fundamental concept in mathematics, particularly in linear algebra, defined as an array of numbers arranged in rows and columns. In the context of the video, the matrix serves as the primary subject, with the script explaining how to identify and manipulate matrices. For example, the script describes a matrix with the numbers '2 7, -4 6, 3 5' and explains that it is a 'two by three matrix'.

💡Order of a Matrix

The 'order' of a matrix refers to its dimensions, specifically the number of rows and columns it contains. This is a key concept in the video, as understanding the order is essential for matrix operations. The script uses the term when explaining that a matrix with 'two rows and three columns' is a 'two by three matrix'.

💡Element

In the context of matrices, an 'element' refers to an individual number within the matrix, located at a specific row and column. The script introduces this term by asking viewers to identify elements like 'element 2,3' or 'a23', demonstrating how to locate and refer to specific numbers within a matrix.

💡Row

A 'row' in a matrix is a horizontal set of elements. The script explains that rows are used to determine the order of a matrix, with the first row being the topmost and subsequent rows below it. For instance, the script mentions 'this is the first row, and this is the second row' when describing the matrix.

💡Column

A 'column' in a matrix is a vertical set of elements. The script discusses columns in the context of determining the order of a matrix and identifying elements. It states 'the columns are vertical, this is the first column, second, third', illustrating how columns are organized.

💡Square Matrix

A 'square matrix' is a special type of matrix where the number of rows equals the number of columns. The script identifies square matrices by stating that they have the same number of rows and columns, like the '2 by 2 matrix' and the 'three by three matrix'.

💡Addition of Matrices

The 'addition of matrices' is an operation where corresponding elements of two matrices are summed. The video script explains this by providing an example of adding two matrices, 'matrix a' and 'matrix b', by adding their corresponding elements, resulting in a new matrix with the summed values.

💡Multiplication of a Matrix

The 'multiplication of a matrix' can refer to different operations, but in the video, it specifically refers to scaling a matrix by a scalar, where each element of the matrix is multiplied by a given number. The script demonstrates this with '4 times a', resulting in each element of matrix 'a' being multiplied by 4.

💡Subtraction of Matrices

The 'subtraction of matrices' is similar to addition but involves subtracting corresponding elements of one matrix from another. The script explains this process by showing the subtraction of 'matrix b' from 'matrix a', resulting in a new matrix with the differences of the corresponding elements.

💡Pre-Calculus

Although not explicitly defined in the script, 'pre-calculus' is the broader mathematical context in which the video's content is situated. It is a course of study that typically precedes calculus and includes topics such as matrices, which are discussed in the video. The script mentions 'pre-calc' in the closing remarks, suggesting further resources for learning.

Highlights

A matrix is defined as an array of numbers organized into rows and columns.

The order of a matrix is described by the number of rows followed by the number of columns.

Identifying a specific element in a matrix is done using row and column indices, such as element a23.

Matrix A is a 2x3 matrix with the given example elements.

Element a12 is identified as the value in the first row, second column, which is 7.

Element a21 is in the second row, first column, with a value of 6.

Matrix B is a 3x4 matrix with its elements provided as an example.

Elements b11, b23, b14, and b34 are identified with their respective values in Matrix B.

Matrix C is a 2x2 square matrix with equal numbers of rows and columns.

Matrix D is a 3x2 matrix with three rows and two columns.

Matrix E is a 1x1 matrix containing a single element.

Matrix H is a 2x4 matrix with two rows and four columns.

Matrix F is a 1x4 matrix with one row and four columns.

Matrix G is a 3x3 square matrix, another example of equal rows and columns.

To add two matrices, corresponding elements must be added together, and the matrices must have the same dimensions.

The sum of Matrix A and B is calculated by adding their corresponding elements.

Multiplying a matrix by a scalar, such as 4 times Matrix A, involves multiplying each element by the scalar.

Subtracting one matrix from another, such as A minus B, is done by subtracting corresponding elements.

The video concludes with resources for further learning on pre-calculus in the description section.

Transcripts

in this video we're going to focus on

matrices

now what exactly is a matrix

a matrix is simply an array of numbers

organized into

rows and columns

and first let's talk about the order of

a matrix

so consider matrix a

and let's say

it has the numbers

2 7

negative 4 6

3 and 5.

what is the order of this matrix

now this matrix has

two rows

and three columns

now the rows are horizontal so this is

the first row

and this is the second row

the columns are vertical

this is the first column second

third

so there's two rows and three columns

so this is considered

a two by three matrix

the order of the matrix

list the rows first and then the number

of columns

now you need to be able to identify a

specific element

in a matrix for example

what is

element

2 comma 3

or sometimes it can be written as just

element a23

so this is in matrix a

and

the first letter represents the row the

second letter is the column

so this is the first row this is the

second row

this is the first column second column

third column

so this element number five is in the

second row third column

so element a two three has a value of

five

now here's another one for you

what's element

a one two

and also element

a two one

go ahead and identify the value of these

elements

so element a one two that's in the first

row second column

so that has a value of seven element a21

is in the second row first column

and so that has a value of six

let's consider another matrix so let's

say if we have matrix b

and it has the numbers

four three

seven negative two

five six negative four

nine

negative three

eight

one

and negative seven

what is the order of the matrix

so let's start with the rows this is row

one

row two row three

and then the columns column one

two

three four

so the order of matrix b

it's a three

by four matrix

it has three rows and four columns

now identify

elements

b11

b

2 3

b14

and b

three four go ahead and identify these

four elements

so this one is in the first row first

column

so that has a value of four

element b two three

that is in the second row

third column

so that has a value of negative four

and then element b14 is in the first row

fourth column

so that's equal to negative two

and element b34 is in the third row

fourth column

so it has a value of negative seven

so you need to be able to determine

the order of the matrix

and the value of every or any element in

the matrix

so what i'm going to do at this point is

i'm going to give you a list of matrices

and i want you to identify the order of

each matrix

so let's say if we have matrix c

and it has the numbers 3

negative 5

2

negative 1

and then we have matrix d

which contains the elements

four

five

negative two

seven

three and negative

six then we have matrix e

which contains one number which is eight

matrix f

it's going to have seven

four

negative five

and

11.

and let's say

matrix g

has the numbers 3 1 7

2 six negative four

nine zero three

so identify

the order

of each matrix

go ahead and try that

let me just give you one more matrix h

which is going to be 2 1

7

negative 3

6 negative 2 5

and 4.

and also determine which of these

matrices

represents a square matrix

so let's start with matrix c

it has two rows and it has two columns

so therefore

it's a 2 by 2 matrix

now

this is a square matrix because

the number of rows and columns are the

same

in the square all sides are the same

now for matrix d

we have three rows

and two columns

so this is going to be considered a

three by two

matrix

so the order of the matrix is always

going to be

the number of rows times the number of

columns

now for matrix e it has one row

and one column

so because it only has one number it's a

one by one

matrix

now for h

there are two rows

and there's

four columns

so this is a two by four

matrix

and then for f

we have one row

and

there's four columns so that's a one by

four matrix

and finally the last one g that's

another square matrix

as we can see there's three rows

and it has three columns

so that's a three by three

matrix so now you know how to determine

the order of the matrix and you also

know how to

identify the elements within a matrix

now let's focus on adding

matrices

so let's say if we have matrix a

and it has the numbers 2 3

5 negative 4

and we have matrix b

which is uh

7

4

negative 3 and

5.

what is the sum

of matrix a and b

so if we add in those two matrices

all we need to do

is

add the corresponding elements

and by the way

if you have a two by two matrix

you can only add it to another two by

two matrix

the number of rows and columns must be

the same when adding matrices or

subtract the matrices as well

so the first element

the one in the first row first column

we need to add it

with uh

element b11 in the first row in the

first column they have to match

so this is going to be 2 plus 7

and then we need to add

these two

so in the first row second column is

going to be 3 plus 4

and then we're going to add those two

numbers

so that's going to be five

plus negative three

and these two numbers which are in the

second row second column

the result will remain in that position

second row second column

now two plus seven

is nine

three plus four

is seven

five plus negative three is two

negative four plus five is one

so this is the sum of matrix a and b

and that's a simple way to add two

matrices together

it's not very complicated

now let's say if you want to multiply

matrix a by four

what will you get

four times a all you need to do is

multiply every element by four

so it's going to be four times two

four times three

four times five

and four times negative four

so then 4a

is going to equal

8

12

20

and negative 16.

so what about subtracting two matrices

let's subtract matrix a and b

so we're going to start with a and then

subtract it by b so it's going to be 2

minus 7

3 minus 4

5 minus

negative 3

and then negative 4 minus 5.

so a minus b that's going to be two

minus seven which is negative five

three minus four is negative one

five minus negative three or five plus

three that's eight

negative four minus five is negative

nine

and so that's the difference

between

the two matrices

and that's it for this video if you want

to find more videos on pre-cal

feel free to check the description

section of this video i'm gonna post

some links there so thanks again for

watching

you