Matematika kelas XI - Matriks part 1 - Ordo dan Dasar Operasi Matriks

Well, this is horizontal and vertical 1*1 both of them

have no friends. Well, this is what failed.

It failed immediately.

[Music]

Hello kids, welcome to the

BJ Kos channel. I'm Koben. In

this video, Koben will discuss

matrices, okay? In this matrix, Koko

will divide it into several parts, okay? In the

first part, we will talk about the order, okay?

Order. So, what is meant by

matrix order? Yes, the matrix is ​​an order, the

term is rows times columns. Rows

times columns, yeah. What is that line? The row is

horizontal, the column is vertical. Well, so

this order is rows times columns. Well, the order is

an example, for example, there is an order of 2 * 1. So,

what does that mean? There are two rows,

one column. Well, for example,

AB is the letter AB, right? So there are

two rows. So there are two horizontal ones

. This row is two columns one. Well,

that's the order of things. Row times column order. So that's how it is

. Well, if you fill it with numbers,

you can do it. So, let's say 31.

Well, this is the order of 2 * 1. Let's say the

order is

1 * 2. This is the reverse nail, yes. Well, 1 * 2

means yes, there is 1 row, there are two columns.

This is A, this is B. So there is one row,

two columns, right? So this is mass, right?

It's not close, this is mass, the term is column

one, column two, right? So, order 1 * 2.

So, the matrix looks like this, there are

boxes like curly brackets

. Well, this is called a matrix, right?

Consists of rows times columns.

For example, Koko will give an example,

he wrote 31. Well, this is the order 1 * 2.

For example, the order

2 * 2. Well, if the order * 2 * 2 is A B C

D. Well, like this. This row has two, column has

two. Well, here are two rows,

two columns. For example, if we give an example,

let's say the numbers are 1 2 3 4. Well, this is

the order. Another example, let's

say the order is 3 * 2. Well, the order 3 * 2 means

there are 3 rows. Well, this is the definition of order

first so you understand. This is 1 2 3 4 5 6.

Well, this is how it is. So there are three rows.

There are three rows, two columns.

Well, this is the order. This is called order.

Well, these are the kids. Why is there an order

where if the order of the rows and

columns is the same, this is said to be a

square matrix

. So this is the type of matrix it is a

square matrix.

So, these are the kids. So if the rows and

columns are the same, it is called a dead

square. So, the

sides of a square are the same. So the term is 2 * 2.

Hey, for example, if the order is 3 * 3,

is that also called a square matrix

? Yes, for example, why give

the order 3 * 3 as an example.

So, it's 1 2 3 4 5 6 7 8 9. So, this is

this odo 3 * 3. This is also called a

square matrix because the multiplication of

rows and columns is the same

. This is also called a square matrix

. Well, here, Koko, there is an

addition, there is a square matrix.

This kid, there is also something called an

identity matrix, right? Identity matrix.

Well, this identity matrix exists in

this square matrix,

the identity matrix must have the same order. But

the meaning of the term

matrix identity is that the diagonal has a

value of one. This is a must. So the

diagonal of the identity matrix

must be one. Well, if the order is 2 * 2 that

you explained earlier, it's up to you what

number you want, it's

up to you whether it's a square matrix or whatever number you want

. Yes, this is just an

example of Koko's 31. You can

fill it with 1, 2 or whatever you want

. Want min -3 -1, up to you. Well, but

if the identity matrix is ​​a

square matrix. For the order 2

* 2, the diagonal that is slanted to the right is

this one down to the right, this one is 0. That is the

identity matrix, yes, its symbol is called the

identity i. This is a 2*2 identity.

What if the identity is 3 * 3? Well,

this is a 3 * 3 identity, right? So it will be

1 0 1 0 0 1. Well, this is the identity matrix

for 3 * 3, right? So the diagonal is slanted to

the right by 1, right? The others are 0. If

you guys were talking about the order, for example the

order, for example the order is 2 * 2, the

elements of the matrix

can be written as A11,

A12,

A21,

A22. So what do you mean? So this

is the value of the matrices in the matrix

which is located in row 1 column 1, right?

So that's the meaning. So A11,

A12 means where the value is.

Column row column row column row.

So row 1 column 1, row 1 column 2,

row 2 column 1, row 2 column 2. So

for example there are values ​​3 4 5 6. Well, that means the

value 3 is located in row 1 column

1. The value 4 is located in row 1 column

2. The value 5 row 2 column 1. The value 6

row 2 column 2. So it is called a

matrix element in a matrix of order 2 * 2.

The values ​​are 3 4 5 6, right? That means

A112 21 22. If you read in the book, it

means that the matrix elements have values

that are located in the rows and columns.

Yes. Second, let's get into

matrix calculations. We

calculate. So matrix calculations

involve addition, subtraction, and

multiplication, right? There is no division.

The matrix cannot be divided. So

first we'll talk about addition,

calculations in matrices. Well,

the addition of the numbers that are added

must have the same order. the matrix order

must be the same. The order of the matrices must be

the same,

meaning K. So, if the order of

the matrices is not the same, they cannot be

added. Because of what? Because it's a

direct example, yes. This is an example of

Kukup having a matrix whose

values ​​are, for example, 2 -1 3 4

+ 0 ee 3 2 -1. Well, if we add

these, the matrix orders must be the same. Well,

this order is 2 * 2, the order is also 2 * 2.

Well, this means it can be added. Because of

what? Because the addition must be in the

same row and column position. The

matrix elements must be in the same position.

So, for example, 2 + 0 means 2. Let's just

write it. Write 2 + 0 -1 + 3. So

the position that is added is the same as 3 + 2 4 + -1,

just 4 -1, the result

is 2

5 3. Okay, this is how to add matrices.

So the positions that are added up must be

the same, so the order must be the same. Why?

If, for example, the order is not

the same, then there will be

different rows and columns that

cannot be added together with the others because

their positions are not the same. That's why the

order must be the same.

Secondly, we go into the

reduction calculation, OK? The subtraction of

these matrices must also be of the same order. The order of the matrices

must be the same. So if it's not the same, it

can't be done. So the theory is the same as

addition, right?

Hodo matrices must be the same. Suppose this is the

wrong matrix. 3 4 - 0 3 2 -1.

Well, this is reduced, right? But, well, the location

must be the same as the one we reduced. 2 - 0

-1 - 3

- 2

4 - -1. Just write this, okay? 4 -1

clearly means the result is 2 -4 1 5 yes,

this is the result of the subtraction, bro, give

another example of subtraction, so for example,

3 -1 2 0 1 3

2 0 -1 1 3

2 like this, okay? Well, this is the order.

This is the order of 3 * 2 minus this order is

also 3 * 2. So this can be

done. But if

the order is different, for example the order is 2 * 2,

what will automatically be subtracted from the bottom part by 1? There is no such thing as 2 * 2.

That's why the order must be the same.

So that's what it means.

Koko, just reduce it immediately, okay? Here 3 - 2

1 -1 - 0 -1 2 -1 means 2 + 1 means

this 3 0 - 1 -1 1 - 3 this -2

3 - 2 means 1. Yes, this is the result. This is the

third one, okay?

matrix multiplication calculations.

This matrix multiplication has the condition of ee ordo.

For example, the order is A * B multiplied by the

matrix order B * C. Now, the position of the

middle one must be the same. What this means is that

this cabbage is a row like a column. So

the columns of the first matrix must be equal to

the number of rows in the second matrix. This is

a must, otherwise you won't be able to

multiply it. That is the condition for matrix multiplication.

So later the result will be A * C.

Well, like this. This is the order. The order of the

matrix multiplication is this is missing.

So, A * C. So, the rows and columns are A

and C. Later, this will be an example of

matrix multiplication. This already has a

matrix of 10 -1 3 5 equals 4. So this is

order 2 * 2, yes, multiplied by this order 2 matrix, the

rows are 2, the columns are 1. So, can this be

multiplied? Can. Later the results of

the order mean that we consider this as being

crossed out, okay? So the result of the order will be 2 *

1. Well, here are the results. So, now

how does the

multiplication process work? Well, the multiplication process is

like multiplying the order of rows

times columns. So, rows times columns.

So, rows times columns. So, later it will be 1 *

5.

So, like this. So the number 1 * 5 + amb

is 0 * 4. Well, that's how to multiply it. So

row times column, but multiply

each number by this one. Well, this is it. So

don't do 1*5 then 1*4 again. Don't.

But the replacement is according to the number of

numbers there are, okay? That's why

it has to be the same. Next one.

Now we are like this. If it's already a row,

this column is

finished. This is only one column, so

let's move to row 2, that's what it's called

. So, -1 * 5

+ 3 * 4. Well, like this. So, rows times

columns again. So, rows and columns like that.

The result is 5 + 0 -5, right? -1 * 5,

right? + 12 3 * 4, meaning the result

is 5 then 7. Well, the result is that

there are two rows, one column.

Well, here are the results. This is meant to be

an explanation of the results of the

matrix order, yes. Well, the

multiplication operation is like this. This is the

second example, brothers and sisters. This Koko has an order of 2

* 2 in * order of 2 * 2. So the

final result of the order must be 2 * 2. So

why is this? This seems to be

lost, right? This Koko, just circle it. Well, this is what

it means. the results of the order.

Well, now the way to multiply it is the same as this

. Well, now rows times

columns. So, row 1 column 1. Well,

first multiply 1 * 2

plus it's like that. Then add 0

* -1

so it's like that. So 1 * 2 + 0 *

-1. Well, now there is column two.

This means we stay on row one but

appear in column du. So, dig it

here and then put it in column 2. Koko,

delete this first so it's enough. So 1 * 4

plus 0 * -2.

Well, this is how it is. So row column 1 * 2 + 0

* -1. Then row column again 1 * 4 + 0 *

-2. Well, that's it. Well, the problem is that there are

no more columns. There are

only two columns, okay? Later these two columns will

be closed. So the column follows the

right one, the multiplication is the right one.

The line follows the left one. If there are

two rows then two rows means

if there are two columns then two columns. Later

this one is now a column row again.

Now -1 * 2. -1 * 2 + 3 * -1 like that

.

So the column row becomes -1 * 2 + 3 * -1

so it adds like that. Then move the column,

now the column is finished, the

other column is in row two rows

times another column. -1 * 4

plus yes.

3 * -2.

Well, like this. If you guys look at this

, the front is the same as this -13 -13 is the same, right?

This is 10.

Well, but this one is the same as this one, the top and bottom are in the same

position, right? This result

is

2 + 0 yes.

2 + 0 4 + 0 so 0 times that is 0 yes, want to

multiply by -2 want to multiply by -1 yes this is -1 * 2 -2

+ -3

this is -4

at -6 yes this is 3 * -2 okay so

the result is 2 -5 4 -10 now this

matrix becomes 2 * 2 this is the result yes

that's what it means.

Here is another example of multiplication,

okay? This is row 3, column 1.

So this is 3 * 1 times row 1. This is

column 3, right? So each column

only has one value, right? So

this is 1 * 3 row 1 column is 3, meaning the

resulting order will be 3 * 3. Well, that's the

final result of the odonya. Do I need to

write this? It's not necessary

actually. This Koko is just

explaining. The important thing is that you can

calculate the multiplication. That's what

matters. This problem is just an

explanation that oh yes this can be

multiplied. Why? Because the columns and

rows are the same, right? Now Koko,

multiply

this row by column first. So -4 * 0.

Well, because this is only one value,

automatically there are no

additions like this, right? This is

there because there is another -1

below it. There's nothing like this, just 0

, just one value 0. Okay,

now moving columns means rows times

columns again. So -4 * -3. Well, that's it. This is a

different column, not one column

. Well, then there are no additional things to this either

. So,

for example, this time, add more

. Well, because each

row and column has two values, so if

there is an addition of one in one row there are

three values, in one column there are three

values, meaning there are more additions,

right? So that's what it means. -4 * 5,

delete this first, okay? -4 * 5 and this nail closes

because there are no more columns. Well,

now move to row 2. Row 2 column

1. -2 * 0 -2 * -3

-2 * 5.

Well, then move again to row 3 column 1.

1 * 0 1 * -3

1 * 5.

Well, like this. If you look at this, the

column positions are all the same, all the

numbers.

This means the result is 0

-4 * -3 12 -4 * 5 -20

0 KO down first, okay? This is 0 too

because this is indeed 0, the one who

multiplies it. Then ee -2 * -3 6 1 * -3 -3

-2 * 5 -10

1 * 5 well this is the result of the order 3 * 3

here, yes, the rows are 3, the columns are 3. Koko, here

is an example, for example, Koko wants to

multiply this matrix. This multiplication

is a 2 * 2 matrix, this is 1 * 2. Well, this is

what Koku said that if this is not

the same, this cannot be multiplied,

what is the result? K 2 * 2. Can't do this, huh.

Why? Let's give an example, okay? We want to explore,

for example, row times column means 1 * 1,

you know. Later, the second one will have these two numbers

. Who added twice? There are

no friends here. So, if it's like this,

1 * 5, 0 * 4, the rows and

columns will be horizontal and vertical. Well, this is

horizontal and vertical 1 * 1, both of them

have no friends. Well, this is what failed. It

failed immediately. Well, that's what it means.

It's no longer possible because

who wants to multiply these two? There aren't any. Well, this

means we can't multiply it, right? J

if possible yes because this is the

requirement for matrix multiplication. This is the property of

matrix multiplication. This is the property of

matrix multiplication. Why don't you first explain that the

property of multiplication is that if you have

matrix A * matrix B, it is not the same

as matrix B * A, right? This is not

the same. So this matrix multiplication,

brothers and sisters, cannot be reversed, that's

not allowed. So if you want to

make A * B, then A * B. If you want to make

B * A, then B * A. You can't make B * A, that's A

* B. Here's an example. Suppose 1 2 3 4.

This is matrix A, right? The B matrix is ​​0

-1 1 2. Let's just say this. This is the B matrix.

Here's an example, for example, Koko wants to multiply A * B,

namely 1 2 3 4. So A * B. Now, we multiply 0 -1 1 2.

If you look at the matrix, it

can't be reversed. Let's

prove it, okay? 1 * 0

+ 2 * 1. Well, here's another column row. 1 *

-1 + 2 * 2 yeah. 1 * -1 2 * 2 3 * 0 + 4 *

1 3 * -1 + 4 * 2. Well, if we

multiply this 0 + 2, yes. Let's get straight to it,

I have

-1 + 4 3 0 + 4 -3 + 8 5, okay? Well, this is the

result. This is a * b. Well, let's

prove b * a. Well, B * A means that

B is written first. 0 -1 2 times

A is 1 2 3 4. Here is the result 0 * 1 +

-1 * 3. Well, this is how it is. Another row of columns 0

* 2 + -1 * 4. Then another row of columns 1 *

1, 2 * 3, 1 * 2,

2 * 4, OK.

Well, the result is

0 in -3, 0 in -4 1 + 6 7 2 + 8 10. Yes,

this is proven, kids.

This is the result of A * B. B * A is the result.

This means that the values ​​of this matrix are not

the same. So A * B does not = B * A. So

remember, if you have the command A

* B, A * B is the matrix,

don't reverse it. It's not like

regular multiplication, right? If the value of the multiplication of

values ​​is, for example, 6 * 7, then it can be the same

as 7 * 6. But if it is a matrix

, it cannot be reversed. Well, then there is

no division in the matrix, right? So

there is no matrix division. So

the matrix cannot be divided. There

is no A/B, there is none, right? there are

only a few times less. Here, kids,

if you ask for matrix

A², then the matrix is ​​squared. This is a

square matrix, this does not mean that

each of these is squared, right?

But A^ must be made by the younger siblings to a * a.

So it's a matrix times a matrix again. So it

can't be a^ that means

each number is squared. No way.

No way. So the multiplication is done

again. So 1 2 3 4 in* 1 2 3 4. So you

have to dig it up and multiply it. So it ca

n't be squared directly. This

result means 1 * 1 +amb

2 * 3.

Then the column row again is 1 * 2 + 2 * 4, yes.

3 * 1 + 4 * 3

* 2 + 4 * 4 yes column row.

Well, that's it. Well, this result means

1 + 6 7

2 + 8 10 3 + 12 15 6 + 16

22. This is different, right? a * a is the result

. If a^ then it must be a * a, but not

each one can be squared directly.

It's different if this is a square, 1 4 9 16,

the results are not the same. So

the square matrix must be made a * a yes b^ yes

b * b. If a^3 then a * a * a. So

multiply it 3 times. Hey, how do you multiply it if there is a * a

* a?

What was the result of A * a before? Then multiply this a again, then

multiply it again like that, okay?

If this is a multiplication of three matrices,

why are there examples of questions, kids?

Here is an example of a matrix A 2x 3 -2

matrix B 45 2x matrix C 2Y 26 -2x -6

. Well, for example, if the matrix A

* B = C, then the value of 3x^ - Y is well,

first we make A * B. We make 2x 3 -2

2 * B 4 5 2x = C, this is 2y 26 -2x -6.

Well, like this. So, kids,

first multiply this, first multiply 2x

2x * 4

+ 3 * 2, this is row times column, then

row times column again. 2x * 5 + 3 * x.

Then the column row again is -2 * 4 + 2 * 2.

Then the column row again is -2 * 5 + 2 * x.

Well, this is the same as 2y 26 -2x - 6. Yes.

Well, now let's multiply it first, okay?

This is 8x + 6

10x + 3x

-8 + 4

-10 + 2x yes =

2y

26

-2x

-6.

Well, this is the same as the meaning,

then you are asked to calculate

the variables in a matrix, yes, the

position is the same as the one in the same

position. So what it means is, this is the same

as this, like that. So 8x + 6 = 2y. This

can't be done, you can't

get the xy value. Then we make

this one again. This is 10x + 3x, this is

13x, right? It will be the same as this one 26.

So row 1 column 2 must be the same

as row 1 column 2 on the

right side, the left side is the same as the right.

So later 13x = 26. That means we get

2 x. This is possible, can I look for

this one? This can be directly obtained

x, but actually this looks

okay, let's try it, okay? So -8 + 4

= -2x. That means -4 -2x. Can be the

same as x 2, yes, it's up to you. Continue

next. Well, if you

have got the x, you can actually go

straight in here to look for the

substitution y, so to speak. So come in

here, okay? Replace 8x with 2 + 6 = 2y. So

this means 16 + 6

22, right?

So 22 = 2y. That means y is 11. This is

possible. Well, that's it. Hey, how come I

can use this one? Yes.

The results will be the same. This is -10 move + 4. 2x =

4 x is 2. It's up to you. This nail is enough.

In essence, we can get x and y, which is enough,

right? Just two variables x and y, right? If

there is a new z, we will look for it again, okay? Well,

now we are asked for the value of

3x²

- y. Well, we fill in x with 2. This is 3 *

2² - y is 11. So 3 * 4

- 11 12 - 11 1 yes this is the answer. Here

's an example, Koko. Next, here is

the matrix A 32 -3y -10X B -15 -4y -2y C.

x1 -3 3 -4 -3x d 1 2y -2y -1 yes. If a

+ b = c * d determine 4x + 3y. Well, here

we write the matrix a + b directly

. 32

-3y -10x

+ -15

-4y

-2y

c is x1

-3 3 -4 -3x

multiplied by

1 2y -2y

-1.

Well, this one. Well, now let's operate it, okay?

Plus.

So, add this, yes, 3 + -1 2, + 5 7.

So, the one that is placed is added, yes,

as I taught you earlier, yes. Then

this is -3y + -4y -7y. Just go straight to it.

-7 -10x + -2y. Well, there's no need to

add this up, just write it down.

Yes, because the variables are different, X and Y. Now, it's the

same as Now,

first multiply the row times the column row column x * 1

+ 1 * y

+ -3 * -1.

Well, this is how it is. Koko wrote it directly so that

later the younger siblings can try it themselves. Let

's try multiplying and then match

the results, okay? This is row times column x

* 2

1 * -2y

-3 * 1. Well, this is how it is. Then we move the

line, okay? The column is finished. This is

moving rows. 3 * 1

-4 * y -4y

-3x * -1 + 3x. Well, this one. Then

another column row. 3 * 2 6 -4 * -2y

+ 8y -3x * 1.

So, this is the addition and

multiplication. We match between section 1

and the left and right sections. This is our

place which is located in the same place as the one above

. Koko, just take this one, okay? These

numbers are clearly 2 and 7. So

later 2 = x + y + 3. Well, let's

solve this first. This means 2 - 3 = x +

y. So x + y is -1, right? 2 - 3 is

-1. This is x + y = -1, right? We can

consider our equation 1 which is 7, make it

7 = 2x - 2y -3, yes, move

this side 7 + 3 2x - 2y, this is 10, yes,

let's divide it by 2 first, you can

divide all of this by 2, this is 5x - y,

now, from this to this, we eliminate.

Elimination, you tap here, write x + y

= -1

x - y = 5, yes. Well, first we'll eliminate it, then

we'll add it so that the y disappears by

elimination, right? Elimination is

removing variables. This is a plus

and a minus, it can be added if you want to

remove it. So later x + x 2x = -1 +

5 4 gets x 2. Now, if x

has got 2, let's just go into this one,

okay? Substitute here and you will get

this. This means x + y is -1. Here, we

fill in x with 2.

Well, this can mean that y is

2, so it moves to -2. -1 -2.

Y is -3. Well, I got it. Oh,

this K isn't done either, why is it the same

as this one? Well, because Koko has already got

X and Y and the variables are only X

and Y. That's enough, kids, you don't need to

do this anymore, it's okay, or

this one is also the same as this one, you don't need to do it, okay?

If my brothers and sisters wanted to use

this and this, then I

eliminated it with this one. May I?

Yes. It's okay, it's the same. Well,

this determines 4x + 3y, this is the final result,

okay?

Well, here x gets 2. 4 * 2 + y

gets -3.

So, this is 8 - 9 -1 you get. Well, that's the

result. Here, Koko, there is

another principle which is that when multiplying

, multiplying a value or constant

by a matrix.

So what happens if the value is multiplied by the matrix

? So, for example, there are 2 times the

matrix ABC D, for example, this value is

2, meaning the number 2 or just a

constant, the constant k, the value of K,

if this is not a matrix, this is a

constant value, if

we multiply the matrix, we multiply one by one

K * A * B K * C * D. So, multiplied

one by one, each is multiplied by the

constant k. So suppose k is

2 * 1 -13.

Well, if there is a multiplication of values ​​with a

matrix, we multiply the values ​​one by one, the

values ​​are entered one by one, okay? J 2 *

1 2 2 * -2 -2 2 * 0 2 * 3 6 that's it.

This is the multiplication of values ​​by matrices. So

dig it one by one. It's different from matrix

times matrix, right? Row times column if

matrix times matrix. If this is the case, multiply it

one by one. This is the

next video, Koko will discuss the

transpose adjoint, determinant and

inverse. Don't forget to watch it, okay?

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