Hello students, I am Dr. Gajendra Purohit and today I am starting the classes of Engineering Mathematics. Welcome to this class. So, the first topic in Engineering Mathematics is Matrices. It is a part of Linear Algebra. And the topics in Matrices in Engineering Mathematics and in Normal BSc are Rank of the Matrix and Inconsistent and Consistent of the Linear Equation and Eigenvalues and Eigenvector and Kelly-Milton Theorem.
So now I'm going to teach you first is rank of the matrix. You all know that what is a matrices because 12th class has a chapter on matrices and determinant. So you all know what matrices are and how we get determinant. But in a little basic you know what matrices are. and then we will start with rank of matrix.
So what is a matrix? A matrix is the arrangement of numbers in row and column. So we represent matrix in this way.
A to n, A, Mn, A, M1, A, M2 and A, Mn. This is matrix and it is M into n matrix. Here M represents number of rows and N represents...
in number of column and the second thing is what is the order of the matrix so the m into n is the order of matrix second thing is what is the square matrix so if in any matrix number of row and number of column are equal then this type of matrix is called square matrix like 2 into 2 matrix 3 into 3 matrix 4 into 4 matrix this type of matrix is called square matrix then here we have Types of matrix are asked, what is a row matrix, what is a column matrix, what is a unit matrix. So, I will tell you where and how it is required. Now, let's start, what is a rank of matrix? If we have any matrix and it has 3 rows, 3 rows and all 3 rows are different. In the sense, no element is repeating or not the same.
So, in that case, if we take out its determinant, then it is... So, the rank of SXS is 3. If there is a matrix, its rows are 2, 1, and JC. If 2, 1, and JC are there, then its rank will be 2. Because, Because here rank means number of different rows. If in any matrix, the number of rows is same, then what will be its rank?
So we will take an example of 3x3. So if there are 3 rows in 3x3 matrix and all 3 rows are different, then its rank will be 3. If there are 2 lines similar to 1, then its rank will be 2. If all 3 lines are similar to 1, then its rank will be 1. So we will talk about definition here. So what is definition of rank of matrix?
Let A be any matrix. A number R is called rank of matrix of A. There are two conditions here. First condition is there is at least one minor of A of order R which does not vanish. And second condition is every minor of A of order higher than R vanish.
So this is the definition and students have a big problem in understanding the definition. So I will explain to you by example what this definition means. So let's take an example and try to understand this definition from that example. Let's try to understand this definition from that example.
What is this definition? So I will explain to you by example what this definition means. Let me take an example from here. Let's understand from that example. You have a matrix.
1, 2, 3, 2, 4, 8 and 2, 4, 6. Let me change it a little. Let me take its number. So we have a matrix here. And if you are asked the rank of this matrix, what is the rank of the matrix?
So what we will do here is, what does minor mean? A minor word is used in this definition. So we have one minor of a order which are which does not vanish or which does not vanish.
What does this mean? So this means that this is a matrix. The sub-matrix of this matrix, what does sub-matrix mean? This can become a matrix, this can become a matrix, this can become a matrix, this can become a matrix. So many sub-matrix can be formed from one matrix.
So the possible sub-matrix that I have told you here, so this big matrix, the sub-matrix of this big matrix, what do we call it? We call it minor. So if I find the determinant of this matrix, which is this small matrix, then we see here, matrix is 7, 2, 5, 4. This is its minor.
So, we will see its determinant will be 28 minus 10. You are seeing which is not equal to 0. So, here we will see this matrix. So, we will see this matrix. So, 5 and 6, 30 and 28 that is again 2 which is not equal to 0. You are seeing this 5. Minus 14 which is minus 9 which is not equal to 0. 16 minus 15. Here you can see the difference is 1. So this is not 0. So you can see in this matrix, all its sub-matrixes that is called minor.
The determinant of all these is non-zero. Now I will talk about this matrix. This sub-matrix is 2 by.
two matrix. Here I took an example, which I had told you, that I would like to explain the definition of rank of matrix, with this example. So see, in this example, you see its sub-matrix, so here sub-matrix 1725, you are seeing its determinant is non-zero.
So M2, we took 2538, its determinant is also non-zero. M3, we took sub-matrix, that is minor, its determinant is also non-zero. M4, we took its determinant, its determinant is non-zero.
So these all matrix are, sub-matrix are 2 by 2 matrix and determinant of all this 2 by 2 matrix is non-zero but you see if you take 3 by 3 matrix so 3 by 3 matrix which is its entire matrix if we take this determinant then what is coming? Zero is coming. So definition says that there is at least one minor of A of order R which does not vanish, means it should be non-zero, means its value should be non-zero, and every minor of A of order higher than R vanish, means here it is coming, that the bigger one, if the determinant of 2 by 2 matrix is non-zero of any one, then the determinant of just bigger matrix should be 0, so you are seeing here, that the rank of this matrix is 2, and you, You can see it directly by chance if you want to use short trick. So in this matrix, these two lines are similar because this is double.
So these two are similar. So ultimately, we will count them as one. And this line is different.
So we are seeing it directly. So what is the rank of this matrix? It is 2. And by calculation and by the definition, we can understand this example. What is the rank of matrix? We will take out the rank here.
I will give 3-4 examples, different examples, how to take out the rank of the matrix. And in this exam, questions are also asked how to take out the rank of the matrix. So I am taking an example in front of you, how to take out the rank of the matrix.
So if students face a big problem, here are 2-3 ways by which we can take out the rank of the matrix. In which I had just told you an example, that is determinant method or minor method, in which we take out the rank of all the matrix. But this method is not very useful because sometimes some methods are not very useful. matrix is not square and we have problem in determining it so in such cases we use matrix and row transformation to find rank. So students find that process very tough that sir we have problem in finding rank and we have to transform it.
transformation, how to know that we have to apply this and that. So I will teach you a trick on how to apply matrix transformation so that our rank can be quickly removed. So I will teach you a small trick. If any matrix is 3x3 or 3x4, like this matrix is given to us of 3x4 order.
So this matrix is of 3x4 order. So first of all our objective will be to make these two zero. This is the help.
So it's a first step and then what will we do in the second step? In the second step these two are zero. What will we do in the next step? We will make this zero with the help of this. If we do these two steps, then your question is solved in two steps.
There is no need to waste time. So how will we do it? Here you see, we will make this and this zero with the help of this element.
So you are seeing, this is row number 1, this is row number 2, this is row number 3. So you take row number 2. In row number 2, what will you do with row number 1? Minus it. As soon as you minus it, what will become here? Zero will become.
In the same way, row number 3 is there. In row number 3, you are seeing that it is 2. So we will make it 2. multiply it and what will we do? We will do minus. So what will happen to us? It will become 0. How?
Look here. The next step of this will be 1, 4, 3, 2. If I want to solve this step, then how will I do it? What is rho 2? 1, 2, 3, 4. And rho 1? 1, minus 4, minus 3 and minus 2. So if you solve it, then here the value will be 0, minus 2, 0 and 2. Then next transformation you will apply this one.
So how will you apply this one? So, we have to multiply R1 by 2. So, minus 2, minus 8, minus 6 and minus 4. So, what is the value of this one? It is 0, minus 2 and what is the value of this one?
It is 1 and 1. This was our first step which I had told you that we made these two 0 by the help of this element. Then, our next step is... Now we have to make this zero by the help of this element. So in front of you, I have given you that we have to make this R3 row zero. With the help of R2.
So what will we do? We will directly minus R2 from this. What will we do?
See, 1, 4, 3, 2. 0, minus 2, 0 and 2. You see here, I am writing this row 0, minus 2, 1, 1. And here if we minus this, then minus 0. minus minus plus minus zero and minus two so what is value of this one is zero zero one and minus one you can see that in this matrix this line is not zero it means that the rank of this matrix will be three this is called ashelem form so with the help of ashelem form we can get the rank of any matrix and what will be the rank of that matrix here If we had done transformation and it would have been 2, 0, then the rank of this matrix would have been 2. But it is not like that. Here it is 0, here it is 1 and here it is minus 1. So the rank of this matrix is 3. Students, here I am taking the second example. And the second example is that same 2-SAM, which I gave you the first example of 3x4 matrix. Again, 3x4 matrix and second example. If you want to find the rank of this matrix, then first of all, we will make these two 0 by the help of this element.
You see, what will happen? R2 minus 3R1. If we multiply by 3 and minus, then it will become 0. Similarly, if we talk about R3, So R3 minus R1 is finished. We will apply these two transformations and it will become 2 0. So how will we do it?
See here. 1, 3, 4, 3. And then R2 is 3, 9, 12, 3. And multiply it by 3. Minus 3, minus 9, minus 12, minus 9. So value of this one is 0, 0, 0 and minus 6. And then second one, see here. See here.
I want to tell you. If you make 2 zeros here, then 0 will be made here and 0 will be made here. So, you... then you have to make this zero with the help of this. Is it clear?
So you will see, here we will make this zero with the help of this. Because you have to... If we want to make 0 here, then we can do it with the help of this.
But here, 0 is already made. So, if we look here, 0 is already made here. Then, we will make 0 here by the help of this element.
So, see what will happen to our transformation? R3. What will I multiply in this R3?
If it becomes 0 when it is directly minus, then you will see, if I multiply it with 3, then it will become minus 6. And minus 6, minus 6, what will happen? 0. So, what will we have to do? We will have to multiply it with 3 and minus R2. So, see what we will do. We have, next step.
our matrix will be 1, 3, 4, 3, 0, 0, 0, minus x, 0, 0, 0 and 0. So this is rank of. So you can see here that the last line has become 0 and now we cannot make 0 anywhere. So the rank of this matrix is 2. Sometimes in exam questions come that find the rank of matrix by reducing it in normal form.
So reducing it in normal form is like this. How to find the rank of a matrix? You have to ask the question to the end and you will get an idea of the rank of the matrix. Now I will tell you a trick because ultimately students should know what to do.
If your objective is clear that you have to reach this point, then you will reach it. But the concept should also be clear. So I want to tell you that this matrix is solved. We got the answer and the answer is the rank. is 2. But question is find the rank of matrix by reducing it in normal form.
So what we have to do is reduce it in normal form. Now what is normal form? So I will explain by example what is the normal form and how we will reduce it.
So I want to tell you if you If your question is asked in normal form, then you will know that it is rank 2, so you can remove it or not write it. Now we have to do that if this matrix is rank 2, then you have to reach here. How to reach here? It means that 2 by 2 unit matrix will be made, rest of the elements will be 0. You have to reach here.
Now you will say, sir, how will we know that we have to reach here? So I want to tell you that you have to know that this matrix is rank 2, it means 2 by 2 unit matrix will be made, rest of the elements will be 0. Okay? So, if the answer is rank 3, then what we do is, if it is rank 3, then the unit matrix of 3 by 3 is formed and the rest of the elements are 0. So, what is the meaning of normal form? To make unit matrix.
What we have to do for the rank of the above and above? To make unit matrix. Now, we were in the example, what is 0, 0 here? We know that the rank of this matrix is 2, which means we have to make 2 by 2. So, you know that your objective is clear that you have to go from here to there. Now how to reach?
So, here we will apply column transformation. Now we have told you about row transformation. Now when we know that its rank is this much and we have to reach here, then we will apply column transformation. Now how to apply column transformation? It is very easy.
What you have to do is to make zero here and here with its help. Now I had told you two minutes ago that if we want to find the rank of any matrix, then we have to do this. Next step is to make zero with scale Next step is to find rank of matrix by reducing in normal form Next step is to make zero with scale What we are going to do is, we are going to make these three 0 with the help of this. So, what is this column number? It is 2. So, in column number 2, multiply 3 from column number 1 and make it minus.
It will become 0. In the same way, if you are looking at column number 3, then minus 4C1. column number 4 so see 4 minus 3 C 1 so if you will apply this transformation then your work will be done very easily 1 0 0 if you will multiply by 3 and minus then you will see 0 0 if you will multiply by 4 and minus then this is 0 0 if you will multiply by 3 and minus then this is 0 0 multiply by minus 0 minus 6 0 clear so your work is almost done here minus 6 is there so make it 1 and interchange it here so what we will do first column 2 is interchanged by column 4 now you will be thinking how will we know to interchange column 2 with column 4 so I want to tell you I want to make 2 by 2 unit matrix and I want 1 here and when will 1 come here when 6 comes here so what we will do is we will interchange this minus 6 so you will see in next step our value What will be 1, 1, 0, 0, 0, minus 6, 0, and 0, 0, 0, and 0, 0, 0. Now you know that if you want to make 1 here, then you don't have to do anything. You have to divide minus 6 in C2.
As soon as you divide minus 6, you will reach near your goal. 0, 0, and 0, 1, 0, 0, 0, 0, 0, 0. And this is the normal form of the matrix. We can write it as i2 0 0 0 0 0. So the rank of this matrix is equal to 2. So if we want to find the rank of any matrix, we can reduce it in normal form and find it like this.
So students, we have just seen two examples of 3x4 and 3x4 matrix. Now, if there is an example of 4x4 matrix and if we want to find its rank, how will we find it? So here we will use the same process which we used in 3x4 matrix. Still, let me explain it to you in a little bit of detail. So you will see here, if we are given 4 by 4 matrix and if we take out its rank, then what will we do?
First of all, we will make these three 0 with the help of this. Then what will be our next step here? Our next step will be here.
We will make these two zero with the help of this. Then, the last step will be this. We will make this zero with the help of this. So, we will use these three steps to get the rank of the matrix.
But, yes, if we are asked in the exam, to find the rank of matrix by reducing it in normal form, then we will have to do some steps ahead. There we will have to use column transformation. So, look, but you are seeing in this question that here is 6. And this 6 can create some problem for us because we will have to do more calculations to apply transformation.
So, we will try to get... column 2 is 1, so we will try to transfer 1 here. So we will interchange column 1 with column 2. So you will see what will happen. Here it will be 1, 2, 3, 4, 6, 4, 10, 16, 3, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, Now what we will do is, as I told you, we will make these three zeroes by the help of this element. What will happen?
Look, R2 R2 minus 2R1, R3, R3 minus 3R1 and R4, R4 minus 6R1. Not 6, it's 4R1. So, we will use this transformation and these three will become 0 after this transformation. So, you will see 1, 6, 3, 8. First, I will put the transformation. How much is R2?
R2 is 2, 4, 6, minus 1, minus 2, minus 12, minus 6, and minus 16. So you can see here its value is 0, minus 8, 0, and minus 17. Next we will put some transformation on this. So as soon as we put it on this, then here also we will have 0, minus 8, 0. minus 17 and then next volume will be 0 minus 8 minus 17 now you are seeing this is our first step then in second step we will make these two 0 by the help of this minus 8 so you are seeing R3 R3 minus R2 and R4 R4 minus R2 so you are seeing here the matrix will be 1 6 3 8 0 minus 8, 0 minus 17, then 0, 0, 0, 0 and 0, 0, 0, 0. So, you can see here that the last two lines are 0. So, we have to write next line. You don't need to do summation.
You got to know that both lines of this matrix are 0. So, how many non-zero lines are left? Only these two. So, rank of this matrix is 2. So, our answer comes here.
But, if in exam, we are asked to find the rank of matrix by the... help by reducing it in normal form so in that case we have to do some steps ahead of it, like I told you after that what we will do here? column transformation so in column transformation we will make these three 0 with the help of this so what we will do?
C2 minus 6C1 C3 C3 minus 3C1 C4 C4 minus 8C1 we will do this transformation so after this transformation what will happen? see here so we will do this After applying this transformation, we will get the value of 1, 0, 0, 0. And as I subtract it from 6, it will become 0. Minus 8, 0, 0. And then again 0, 0, 0, 0. And then if we subtract it from 8, it will become 0. Minus 17, 0, 0. But I had told you that if we know the ring of any matrix, like I told you that how much the ring of this matrix will be, Because here two lines are left, so it's 2. And since its rank is 2, then its normal form will be 1, 0, 0, 1 and the rest of the elements will be 0. So you know that you have to reach here. Now how do we reach here?
Look here. What's happening here is that we have to make 0 here. So how will we make 0 here? We will make it with the help of minus 8. But first look what we need instead of this minus 8. So, we will first do C2 upon minus 8 and C4 divided by minus 17. So, our next step will be 1, 0, 0, 0, 0, 1, 0, 0, 0, 1 and then 0, 0, 0 and then 0, 0, 0. After that, we will make this 0 by the help of this. So, we will put the transformation here.
C4, C4 minus C2. So, as soon as we get the transformation, the answer will be this. And we can write it like this.
I 2 0 0 0 0 0 0 0 0. So, the rank of this matrix is 2. So, this is how we get the rank of the matrix of 4 by 4. I will take one more example and the last example of this topic. Then we will start a new topic. Here, you have taken the rank of this matrix. So as you can see in the previous question, we had written 6 and 1 here, so we had interchanged it here.
So here we have written 0, so we will try to get the non-zero element of the corner element there, so that we can quickly process it. So we either interchange this and bring it here or we interchange this and bring it here. So what I will do here is I will interchange column 2 with column number 1. By doing this, it will come here. 1, 0, 1, 1, 0, 1, 3, 1, minus 3, 1, 0, minus 2, minus 1, 1, 2, 0. You can see that we have a non-zero element here. Then I told you what our first step is.
Our first step is to make these three zero by the help of this. Then our next step is to make these two zero by the help of this. Then our last step is to make this zero by the help of this. So what we will do here is, these two are already zero, so there is no need to worry.
We don't need to make it 0, we will make it 0, that is R3, R3 minus R1 and R4, R4 minus R1. So here we will see what will be 1, 0, minus 3, minus 1, you are seeing here already 0 is made, then here we will see it, so you see R3 is 1, 3, 0, 2 and minus 1, minus 0, plus 3, plus 1. So the value of this one is 0, 3. 3 3 and then the next step is r4 is 1 1 minus 2 0 and then minus 1 minus 0 plus 3 and plus 1 so you can see here the value we have is 0 1 1 and then 1 now the next step is to make these two 0 by the help of this element so we will do the same thing again R3, R3 minus 3R2 and R4, R4 minus R2. So you will see here 1, 0, minus 3, minus 1, 0, 1, 1, 1 and then here R3 is 0. 3, 3, 3 and then minus of 0, minus of 3, minus of 3, minus of 3. So here we have value of 0, 0, 0, 0. And then next step is our R4. How much is R4? 0, 1, 1, 1. Minus of 0, minus of 1, minus of 1, minus of 1. So the value of this is 0, 0, 0. So you are seeing here what happened to both lines of last.
It became 0. We had 4 lines, 2 lines became 0. So how much did our rank become? 2. We got to know this. But in the exam, we are asked to find the rank of matrix by reducing it in normal form. So, what we have to do is reduce it in normal form.
So, you know that it has rank 2. So, what will be the normal form of this matrix? You know that the normal form of this matrix will be 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. So, this is its normal form. We have to reach there from here.
So, first of all, what will we put here? We will put column transformation. Means, these two.
and this will be 0 by the help of this. So, this is already 0. Next, we will make this C3 plus 3C1 and C4 plus C1. You can see that I have put plus here.
Normally, I am putting minus in the transformation, but here I am putting plus because this is positive and this is negative. So, ultimately, if we add both, then what will happen? It will become 0. So, you will see that it will become 0 1 0 0 and then 0 1 0 0 0 1 0 0 up next step America already in the no no co 0 by the help of this element the but simple a half care in here pay C 3 C 3 minus C 2 and C 4 C 4 minus C 2 jay same here transformation like I'm a tomorrow answer I got what it naa up a 0 1 0 0 0 0 0 0 and 0 0 0 0 so the rank of this matrix we can write it like this i2 0 0 0 0 0 and 0 0 0 so the rank of this matrix is equal to 2 so students today i took the topic of rank in front of you next the next topic will be consistent and inconsistent of linear equations So, the whole topic is dependent on the concept of rank. I have taken 4 examples in front of you.
So if we revise today's chapter, I gave you an introduction about matrix. After that, I told you the definition of rank. What is the rank? And then I gave you four examples.
So to be clear, the rank of any matrix is the number of different rows in the matrix. If by chance there are 4 by 4 matrix and there are 4 different rows, then what will be its rank? It will be 4. If by chance there are 2 rows like 1, then in that case, its rank will be 3. If there are 3 rows like 1, then in that case, its rank will be 2. And if there are 4 rows like 1, then it will be 3. Thank you for watching this video.
Stay tuned. I will bring more videos on engineering mathematics. Thank you so much for watching.