Matriks Matematika Wajib Kelas 11 Bagian 1 - Pengenalan Matriks

Assalamualaikum Wr. Wb. Meet me again, Denny Hadayani on the MATLAB channel. In this video we will learn matrix material. I will share the matrix material in several separate videos. This is the first part of the video. In this video we will learn the meaning of matrix, ordomatrix, matrix types, transverse matrix, and the combination of two matrices.

The material I convey in this video, you can download the PDF form, the link is in the video description.

Okay, now we will learn the material about matrix. We start from the understanding of matrix first. This is the meaning of matrix. Matrix is a set of numbers, symbols or expressions in the form of rectangles or rectangles arranged according to the lines and columns.

more clearly, pay attention to the following example, the matrix is like this, friends so inside it, there is a set of numbers or symbols, expressions that we arrange into long rectangles or rectangles then limited by a curve like this the curve can be a regular curve or you can also use a curve like this

The expression or the number or symbol that is in this comma is called an element, matrix element or entry. 2104 - 27 is this matrix element. As in the understanding, here it is arranged according to the line and column. You have to understand which one is included in the line and which one is included in the column.

For example, notice this matrix, which is called a line, is a horizontal element, which is flat. 2, 1, 0, this means the first line.

The bottom one is the second row. Remember, the row is the horizontal. While the vertical one is called the column. This is the first column, this is the second column, and this is the third column. It's clear, right? The horizontal row is the horizontal one, and the column is the straight one or the vertical one. Okay, now the matrix notation.

Matrix is stated with capital letters and the elements are stated with capital letters capital letters are big letters, non-capital letters are small letters If matrix A is a matrix A, I, J, this means the element I, J, states the element that is located on the line to I and column to J So the matrix name we use with capital letters and the element we use with non-capital letters, for example

For example, this is matrix A, we see here we use capital letters, this is the matrix name, and these are the elements.

The element on the second row of the third column, we write A23, which means this is the matrix A element. A is small, which means the matrix A element. 23 means the second row of the third column. Don't read 23, guys. Read it one by one.

Okay, it means the second row of the third column, see which second row, the horizontal row, which means the one next to it This is the second row, the third column, this column is the vertical column, the third column, the second row, the third column, which means the one next to it So A23 is negative 8 The element on the third row of column 1, A31

The third column is this column, the first vertical column is this column, which means the value is A31, which is 9.

Now, there is the term "Ordo Matrix" in the matrix there is something called "Ordo" If a matrix A consists of M rows and N columns, then M times N indicates the size or "Ordo" of matrix A. What is called "Ordo" is the size of the matrix that indicates the number of rows and columns, friends.

So you have to understand first which row and which column. So the order is many rows times many columns. Be careful to reverse. Remember, the row first times the column. Example. This is matrix A.

How many rows are there? The horizontal row, the flat one, oh here there are two rows. How many columns are there? There are three columns, the first column, the second column, and the third column. It means that the ordomatrix is 2x3, because there are two rows, three columns, written A23. Once again, be careful not to go back, if 3x2 it means it's different again, the matrix shape is different again.

So the position has to be a row first, many rows first, then times the number of columns. Okay, another example. For example, this is matrix B. How many rows are there? There are three rows. One, the second row, the third row. There is one column. It means,

This order matrix is 3x1 because it has 3 lines and 1 column. Clear? Okay, now we study the types of matrix. I categorize it into two parts. The first matrix type is based on many lines and columns. And the second, later we study the matrix type based on the pattern.

We first study the matrix type based on the number of rows and columns The first is the matrix of rows The matrix of rows is a matrix that has one row So the column can be any number, the important thing is that the number of rows is only one For example, this is a matrix of rows, why? Because there is only one row The column here is 3 For example, this is also a matrix of rows

There are many rows, only one, and there are two columns. So no matter how many columns, the important thing is if the row is only one, it is said to be a matrix of rows. The second is the matrix of columns. On the contrary, if the matrix of columns, there are many columns that are only one. The row can be any number. For example, this matrix A has three rows, but the column is one, so this is said to be a matrix of columns. Matrix B has two rows, one column, this is also the matrix of columns.

Okay, now what is called a matrix in a rectangle. A matrix in a rectangle is a matrix that has a number of rows and columns that are different. For example, this is matrix A. You can see that its shape is in a rectangle. The rows and columns are different. Then matrix B is also the same.

The lines and columns are different, the shape is also rectangular. So it is said as a matrix rectangular. Then the fourth, matrix rectangular. Well, if the matrix is rectangular, many lines and columns are the same. As an example,

This matrix A, the order is 3 times 3, 3 lines, 3 columns. See, this is square, because many lines and columns are the same. Then matrix B, this is the order 2 times 2, the lines and columns are the same. This is also called a square matrix. In the square matrix, there is what is called the main diagonal. The main diagonal is this 11, friends.

This is called the main diagonal, 1, 4, 1. Then the other diagonal, this one here, is called the secondary diagonal or the side diagonal. Let's try matrix B, where is the main diagonal? 1 and 3. The side diagonal or the secondary diagonal? 2 and 0.

The main diagonal, if we sum up the elements, the sum is said to be a trace. For example, what is the trace? 1 + 4 + 1, which means 6. Here the trace is 1 + 3 because the main diagonal is 1 and 3, which means 4. That's the matrix type based on many rows and columns.

Okay, now we learn the types of matrix based on the pattern of the elements. The first, there is what is called matrix 0. Matrix 0 is a matrix where all the elements are zero. So we don't see any order as long as all the elements are zero, it is said as matrix 0. The name uses the letter O capital, as an example.

O = 2x3, this is a matrix of 0 that has an order of 2x3 or 2 rows 3 columns. See, this is the element, all 0. Another example, this is a matrix of 0 that has an order of 2x2, 2 rows 2 columns. The second is what is called a diagonal matrix. The diagonal matrix is a matrix of squares, so the order must be the same, the rows and columns. While the elements on the main diagonal are not all 0.

with other elements 0, for example, this is a diagonal matrix, it must be a matrix per square, now look at the main diagonal, this can't be all zero, this can't be all zero, so if one or two zero, two entries zero, it's no problem, the important thing is that none of them are zero, while the other entries or elements are all zero, so it's said as a diagonal matrix, for example,

Now, friends, look at the main diagonal, 3, 0, 0, this is one of them that is not 0, so this is called a diagonal matrix, the other elements are all 0, right?

The third one is called the identity matrix. This is also the same matrix per square, but the main diagonal must be 1. The other elements must be 0. For example, this is the identity matrix of the 3x3 order. See the main diagonal is 1.

This is the identity matrix of 2x2 order. Let's see the main diagonal. This must be one. All other elements, except the main diagonal, are zero. Therefore, it is said as an identity matrix. Then the fourth, there is what is called a matrix of three axes. Matrix of three axes, if the element below or above the main diagonal, everything is zero. There are two types of matrix of three axes.

The first one is the matrix of the top three angles. It is symbolized or given a name with U capital. The matrix of the top three angles, which is not zero at the top, while the one below the main diagonal is all zero. As an example, you see,

If the top triangle is below the main diagonal, the opposite is if below the main diagonal, everything is zero. This is the main diagonal, below it is zero. So what we make the triangle is not zero. This is the triangle, so this is called the top triangle. The second, the lower triangle matrix is usually given the name L capital.

For example, this is the opposite. If the lower triangle, it means that the one above the main diagonal is 0. For example, this is the main diagonal, see the top part is 0, it means that we made a triangle, that's not the zero, this is the triangle. So this is called the matrix of the lower triangle.

Okay, those are the types of matrix based on the pattern of the elements Okay, now we learn the transpose matrix Simply put, what is said as a transpose matrix is that we alternate between the row elements and the column elements For example, for example here we have matrix A, this is matrix with an order of 3x3 We will look for the transpose

A transpose is symbolized by the bracket T So if there is A bracket T, it means this is a transpose from matrix A The way we swap between the elements of the line and the elements of the column or vice versa, the column with the line is also no problem You see the first line 1, 2, and 3, this is a line We write, if the transpose is written as a column 1, 2, and 3 So the first line we make as the first column

The second line is 3, 4, 5. If the transpose is, we make the second column 3, 4, 5. And the third line, 2, 1, 1, we make the third column. It's clear, we change or change the line into a column. That is called a transpose matrix. Another example, this is matrix A.

2, 1, 4, 1, 4, 5, 4, 5, 3 We will look for the transpose We see the first line, 2, 1, 4, we make this the first column 2, 1, 4 The second line, 1, 4, 5, we make the second column 1, 4, 5 And the third line, 4, 5, 3, we make this the third column

4, 5, 3, this is A transpose or matrix transpose from matrix A if you see, for example, matrix A and A transpose have the same value the elements are the same, 2, 1, 4, 2, 1, 4, 1, 4, 5, 1, 4, 5

453, 453. If this happens, it turns out matrix A is equal to A transpose, this is said to be a symmetric matrix. This is a bit of a typo. Matrix A is called a symmetric matrix, if it turns out A is equal to A transpose. If in the example 1, it's different, right? It means this is not a symmetric matrix. Clear? Now we discuss the equivalence of two matrices.

Matrix is said the same if the order is the same and the elements that are placed are the same, the same value, for example

For example, we have matrix A and matrix B, matrix A is 0.5, 9, while matrix B is sin P/6, 0, 25, 3/2. Matrix A and B are the same. Why? Because the element is placed the same, for example, this half is the element of column 1.

Here, B is the element of column 1, it's the same, sin pi/6, this is sin 30 degrees. Sin 30 degrees is half too. So this value is the same, half. This is the first line of column 2, 0 is the same too. Then this is the second line of column 1, 5. What is the root of 25?

25 is 5 too, right? This is the second column's second line, 9. This is 3 squared, how much? 9 too. So this is A, the same as B, because the value is the same. Okay, now let's try to discuss some examples of training questions. We start from the first question, the question is like this.

The number of diagonal elements of the main diagonal matrix P, this matrix we will find the number of main diagonal elements. What is called the main diagonal is this part, 5, negative 10, then 1. This is the choice.

So we just have to count it, remember the main diagonal is 11, which means 5 + -10 + 1 5 + -10 is -5, -5 + 1 is -4 So the answer is D Let's continue to the second question

The next matrix is the matrix of identity. Which one? Identity is a matrix of a square whose main diagonal is 1. This is not identity. See the main diagonal is 1, 0. B is also not identity. The main diagonal is also 1, 0. C is also not. This is the diagonal on the side, which is 1. So this is not.

D is the matrix of identity because the main diagonal is 1 while the other element is 0. So the answer is D. Now we discuss the third question.

We know that this matrix is a diagonal matrix. The ABC value in order is... This is a diagonal matrix. Remember that the main diagonal matrix is not all zero. While the other elements are zero. So this is another element. Besides the main diagonal, it must be zero. b-1 = 0, so b = 1.

Then, this is just first, a + b must be 0 too, the point is that the non-zero is the main diagonal. So from here we get, oh yes, we just replace b with 1, a + 1 = 0, so a is negative 1, then from here,

2a + c = 0, we get a = -1 2 times a, we replace a = -1 + c = 0 2 times -1 = -2 + c = 0, so c = +2 Here what is asked is the value of abc in order, that is from a first, -1 b = 1, c = 2

Negative 1, 1, and 2. The answer is C. Let's continue to the fourth example. Question number 4. We know the matrix of the upper three corners. This is the matrix of the upper three corners. The value of X that meets. Remember, if the matrix of the upper three corners, it means that the bottom one is zero. This part is zero.

So, y-1 must be 0, so y = 1. This is also the same, must be 0. x-4y = 0, y is already 1. So, x-4 * 1 = 0, x-4 = 0, then what is x? x is positive 4. Is there? The answer is A.

Okay, now we discuss the fifth example, this is the last question that we will discuss in this video. Known matrix A, this is matrix A and this is matrix B. If A = B, then A transpose =

We use the same matrix, remember that the two matrices are said to be the same if the order is the same, this matrix A and B have the same order, namely 3 rows 2 columns, this is also the same 3 rows 2 columns

And the second is said the same if the element that is located is the same, the value is the same. So the first column is 1, this is also the same. This is 4, this is also the same. This is -5, this is also the same. Here what is asked is A transpose. So we have to know first how many elements are there. To get this value, we have to know first the value of A, this is also the same. So our target, we will look for the value of A first.

We use the similarity of two matrices First we use this part first Why this part? Because the variable is only one We get b-4

The first three columns are equal to this one. This is also the same. The first three columns are negative 9. So, what is b? Negative 9 plus 4. Negative 5. Now, we will look for the value a. We use this one.

5a + 1 = 6 - b, we already got b 5a + 1 = 6 - b, we replace it with -5 So 5a + 1 = 6 - -5 = 11 So 5a = 11 - 1 = 10, then what is a? 10 divided by 5, a is 2 So the value of matrix A, the element, we can complete it

2a + 3 = ? a just replace 2 2 x 2 = 4 4 + 3 = 7 5a + 1 = ? a replace 2 5 x 2 = 10 10 + 1 = 11 this is the choice so matrix A is 1 7 4 - 5

Then -9, 11 Now we will look for the A transpose

Remember that the transpose changes the line into a column and the column into a line So the first line, 1, 7, we make the first column 1, 7 The second line, 4,-5, this is the second column 4,-5 and the third line, -9, 11, this is the third column -9, 11, is there? 1, 4,-9, 7,-5, 11, which is D, this is the answer

Okay, until here first video for matrix material, the next video, the second part, Insha'Allah we will learn matrix operations, multiplication, reduction, and multiplication. Assalamualaikum warahmatullahi wabarakatuh.