Intro to Matrices
Summary
TLDRThis 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
📊 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.
🔢 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.
➖ 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.
Mindmap
Keywords
💡Matrix
💡Order of a Matrix
💡Element
💡Row
💡Column
💡Square Matrix
💡Addition of Matrices
💡Multiplication of a Matrix
💡Subtraction of Matrices
💡Pre-Calculus
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
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