Rank of Matrix Using Transformation | Normal Form | in Hindi by GP Sir


Hello students, I’m Dr. Gajendra Purohit and today I’m starting the classes of Engineering mathematics I welcome you all to the classes So the first topic in engineering mathematics is ‘matrices’ Its a part of linear algebra The topics of matrices that come in B.Sc and engineering mathematics are ‘Rank of a matrix’, ‘Inconsistent and consistent linear equations’, ‘Eigon values and eigon vector’ and ‘Cayley Hamilton theorem’ Firstly, I’m going to teach you ‘Rank of a matrix’ You all know, what is a matrix as we have studied a chapter ‘Matrices and determinants’ in class 12th and how to find out a determinant but I’m still going to teach you some basics and then we will start with ‘Rank of a matrix’ (definition) We represent a matrix in this form Here, ‘m’ represents the number of rows whereas ‘n’ represents the number of columns where mxn is the order of the matrix A matrix in which the number of rows is equal to the number of columns is called a ‘Square matrix’ Example: 2X2 matrix, 3X3 matrix, 4X4 matrix these are the examples of a square matrix Types of matrics are also asked like Row matric, column matrix, unit matrix and I will you about these when needed. For now, let’s start If we have a matrix with three different rows with no same elements or no repetitive element then its determinant will be non zero In such cases, the rank of that matrix is ‘3’ if in any case, two rows of such matrix are the same then its rank will be ‘2’ because rank is defined by the number of different rows i.e. the rank of a matrix is equal to the number of rows of a matrix Let’s take the example of a 3X3 matrix if all its rows will be different, then its rank will be ‘3’ if 2 of its rows will be same, then its rank will be ‘2’ if all its rows will be same, then its rank will be ‘1’ (definition) Students usually face a problem in understanding the definition So I’ll teach you this with the help of an example A matrix can be divided into many sub-matrices These sub-matrices are known as ‘Minors’ The determinant of its ‘sub-matrices’ or ‘minors’ is non zero With the help of this example I would like to explain to you the definition of ‘rank of a matrix’ The determinant of all the minors m1, m2, m3, m4 is non zero. All the sub-matrices are 2X2 matrices with non zero determinants. But if we take its 3X3 matrix its determinant is zero Coming back to the definition, it says that (definition) which means it should be non zero (definition) which means that the matrix of just higher-order should have determinant as zero The rank of this matrix is 2 because clearly, using the short trick we can see that two rows of this matrix are same hence both of the rows will be counted as one We will now find the rank of three to four example matrices which are very important from the exam’s point of view So I’m putting an example in front of you to find the rank of the matrix There are two to three methods for finding the rank of a matrix One of which was the previous example ‘Determinant Method’ or ‘Minor Method’ in which we find the rank using sub-matrices But this method not very useful as it cannot help in determining the rank of matrices other than square matrix In such cases, we use ‘Row and Column Transformation’ Students usually find this procedure quite difficult as they find it complex to choose the type of transformation to apply so I’ll tell you a simple trick to find the rank quickly Let me tell you the trick Since the last line in the matrix is still non zero hence, the rank of this matrix is ‘3’ This is known as ‘Echelon form’ and we can find the rank of a matrix using this and here the rank is ‘3’ If in case, we would have got all the elements of last line as zero after applying the transformation then the rank would have been ‘2’ Students, here I’m taking another example of 3X4 matrix You have to find the rank of the matrix The rank of this matrix is ‘2’ Sometimes. in exams, it is asked to find the rank of a matrix by reducing normal form I’ll teach you how to do find rank by reducing it to normal form By looking at this, now you have this idea in mind that the rank of this matrix is ‘2’ but now I’ll tell you the trick because at the end students should know what they have to do If your objective is clear as to where you have to reach, then you’ll reach but your concepts should be clear Although this matrix is solved and we have got the rank but the question is asking us to find it by reducing it in normal form that’s why we will have to reduce it to normal form With the help of an example, I’ll tell you what is a normal form and how can we reduce it If you are asked to reduce in normal form then although you know that the rank is ‘2’ but still don’t write it down directly Now if the rank of this matrix is ‘2’ then we have to reach here by making a unit matrix of 2X2 and making all other elements as zero You have to reach from here to here You must be wondering about how would you get to know where to reach You can determine it using the fact that its rank is ‘2’ and therefore you’ll get 2X2 unit matrix with rest of the elements as zero If in case you get the rank of the matrix as ‘3’ then you would have to reach 3X3 matrix with rest of the elements as zero ‘Normal form’ means to make a unit matrix of the same rank In the current example, we have a matrix of rank ‘2’ hence we have to make a unit matrix of 2X2 We know that our objective is clear, the only question is ‘how’ here we will apply column transformation Previously we studied row transformation now that we know the rank We are applying column transformation So if we have to find the rank of a matrix we can find it by reducing it to normal form like this So, students, we have seen two examples of 3X4 matrix Now we are taking 4X4 matrix and we have to find its rank then how will we find it We are going to use the same process here as well still, I’m explaining you We will use these three steps to find the rank and if we are asked to find it by reducing it to normal form, then we will apply column transformation Here, in this question, you can see the first element is ‘6’ which can create a problem in transformation during the calculation This is how we find the rank of 4X4 matrix I will take one last example before going on to another topic Here you have to find the rank of this matrix In the previous question also, the first element was ‘6’ so we interchanged two columns similarly, here we will interchange to avoid zero as the corner element to make the process fast Since we had 4 lines where 2 of them became zero therefore rank of the matrix is ‘2’ but if in the exam, we would be asked to find it by reducing it to normal form then we will have to reduce it Students, today I picked up the topic ‘Rank of a matrix’ Next topic will be ‘Consistent and Inconsistent linear equations’ That entire topic is dependent on the concept of rank I took four examples in front of you To revise, I gave you the introduction of a matrix, I told you what is a rank and then took four examples to explain In short, the rank of a matrix is the number of different rows in that matrix For a 4X4 matrix with 4 different rows its rank will be ‘4’ for two same rows, its rank will be ‘3’ for three same rows, its rank will be ‘2’ and for all the rows as same, its rank will be ‘1’ Thank you for watching this video and stay tuned Very soon, I’m coming up with more videos on Engineering mathematics Thank you so much for watching

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