Engineering Mathematics 01 | Linear Algebra (Part 01) | GATE 2025 series | All Branch

Summary

Highlights

• Vinay Kumar commences a comprehensive mathematics course for GATE 2025. 🎓
• The course is free and covers the entire GATE syllabus for various branches, focusing on linear algebra first. 📚
• Vinay shares his extensive experience and achievements, establishing trust and credibility. 🏆
• Interactive lessons through Unity app make learning engaging and doubts are easily resolved. 📲
• Detailed explanation and examples of matrices, including types and their properties, are covered. 🔍

Key Takeaways

• Vinay Kumar is back to teaching mathematics for GATE 2025! This course covers all branches and is packed with comprehensive content. 🎉
• This session introduces matrices from the ground up, making it perfect for beginners! 📚
• The course promises to cover the entire GATE syllabus for multiple engineering branches, ensuring you don’t miss out on any crucial topics. 🎯
• Engage actively with the course through interactive tools like Unity, making sure your doubts are addressed and your notes are well-organized. 📝
• Get ready to discover the beauty of mathematics with fun examples and practical applications! 🚀

Overview

Chapters

• 00:30 - 03:30: Introduction and Course Overview The Introduction and Course Overview chapter lays the foundation for the entire course, providing insights into the main objectives and themes. It outlines key topics that will be covered and establishes the learning outcomes expected from participants. The chapter emphasizes the importance of understanding the foundational concepts to successfully engage with the rest of the course material. Participants are encouraged to actively participate and take notes for better comprehension and retention of the discussed topics.
• 03:30 - 04:30: About the Instructor The chapter introduces the instructor, who welcomes the students to the channel and checks the audio and video quality. The instructor ensures everything is working fine before proceeding.
• 04:30 - 11:00: Course Structure and Schedule The chapter begins with a warm welcome to participants of a mathematics class, emphasizing the excitement of continuing education in math after a long break. The instructor acknowledges the absence of a detailed English course for mathematics in the past, particularly during the forgate 2024 series, and expresses enthusiasm about launching a course now. Additionally, the instructor addresses inquiries about advanced math topics, such as those seen in the previous XC series, and notes the demand for a comprehensive mathematics course, highlighting technical subjects like fluid mechanics.
• 11:00 - 16:00: Unity App and Interaction The chapter "Unity App and Interaction" begins with the instructor expressing their happiness to engage with the students in the course. After extensive discussions and meetings, the course on engineering mathematics is now available. The instructor, Vin Kumar, introduces himself and addresses common questions about the mathematics course, encouraging students to attend the class attentively for the first few minutes to get answers and reassurances about the course progression and materials.
• 16:00 - 19:00: Start of the Lecture - Linear Algebra The chapter introduces the start of a comprehensive online mathematics course, specifically focusing on Linear Algebra. The instructor assures students that the course will cover the entire syllabus of mathematics, and mentions that a detailed syllabus will be discussed after a few slides. This course is offered free of charge, and the instructor aims to provide 100% of the course content to students.
• 19:00 - 31:00: Definition and Notation of Matrices The chapter titled 'Definition and Notation of Matrices' begins with an introduction by the speaker, Vumar, who is part of the mechanical team at PW Gate. He establishes credibility by mentioning his high ranking (A4) in the Gate examination within the engineering sciences paper, highlighting the toughness of the mathematics section compared to other branches. Vumar's past experience includes working with prominent organizations such as HPCL and BARC (Baba Atomic Research Center). The introduction is aimed at building trust with the audience.
• 31:00 - 62:00: Types of Matrices The chapter titled 'Types of Matrices' seems to discuss a speaker's personal experiences and qualifications related to teaching and preparing students for competitive exams such as GATE and JEE Advanced. The speaker mentions having qualified eight times for GATE exams and has been involved in teaching related subjects for over 7 years, highlighting a transition from teaching JEE Advanced to focusing on GATE. There is a reference to 'engineering sciences' but the explanation is postponed. The content appears more focused on the speaker's background rather than the types of matrices.
• 62:00 - 84:00: Key Properties of Matrix Operations The chapter titled 'Key Properties of Matrix Operations' seems to stray from its intended topic. Instead, it provides an introduction to the instructor's teaching background and expertise. The instructor has experience in teaching various engineering subjects, including engineering mathematics, fluid mechanics, thermodynamics, and heat transfer, primarily focusing on mechanical engineering. The current discussion highlights that the instructor will be teaching engineering mathematics, offering a complete course on this platform. However, the specific properties of matrix operations are not discussed in the provided transcript.
• 84:00 - 109:00: Symmetric and Skew-Symmetric Matrices The chapter titled 'Symmetric and Skew-Symmetric Matrices' seems to introduce the syllabus for mechanical, civil, ECE, triple computer science, XC, and data science. The speaker emphasizes the responsibility of students to verify which parts of the syllabus are relevant to their specific branch and to focus only on those areas. For example, computer science students should concentrate on linear algebra, probability and statistics, and basic calculus, excluding vector calculus or complex calculus. The context suggests a discussion about course content related to matrices in the upcoming sections.
• 109:00 - 118:30: Examples and Problem Solving The chapter emphasizes the importance of understanding syllabus requirements and encourages students to align video content with these needs, acknowledging that matching content to syllabus demands is the responsibility of the learner. Additionally, the chapter highlights the launch of a dedicated English Telegram channel, providing updates and resources. Students are encouraged to join this channel using a QR code for direct access to English batch notifications.
• 118:30 - 129:00: Conclusion and Next Steps In this chapter titled 'Conclusion and Next Steps,' the focus is on the creation of a Telegram group for students. The group, which already includes over 860 members, is intended for open communication and support among members, including direct interactions with the speaker. It serves as a platform for students to post queries, engage with content related to the Gate Wala English Channel, and is maintained collaboratively by the students and the speaker.

Engineering Mathematics 01 | Linear Algebra (Part 01) | GATE 2025 series | All Branch Transcription

• 00:00 - 00:30 for
• 00:30 - 01:00 so welcome back to this channel students I hope all of you are doing well so please type in the chat box is the audio and video fine to all of you so everyone out there so please type in the chat box is everything fine is the audio and video fine yes so please confirm the audio and video so we shall start very soon good
• 01:00 - 01:30 evening all of you yeah even I'm happy to see all of you back for mathematics again okay because it's been so long actually I think you know in the past forgate 2024 we haven't launched a detailed course in English for maths actually but uh yeah now we got the chance we'll definitely look so first of all many of you have asked me like last year when I was doing XC series okay that advanced Topics in maths or uh when we were talking about fluid mechanics or basically in technical subjects at few points many of you have asked me for mathematics course okay okay and we were
• 01:30 - 02:00 happy to get this course to you so after a lot of discussions and meets finally we we able to meet here for engineering maths again so first of all I would like to clear some general questions of mathematics okay how the how the course progresses yeah you can get the notes I'll just uh just listen to the class for First 5 10 minutes you'll get answers to all your questions good evening all of you yeah nice okay so see here so first of all of course me myself Vin Kumar okay I'm going to teach you mathematics for this complete batch okay okay so firstly the questions is is this
• 02:00 - 02:30 is going to be a complete batch of mathematics or am I going to teach only few topics of maths okay I'm telling you it is f length course okay so I'll briefly give you the syllabus after two three slides so you can actually get 100% of your syllabus from this course okay Paka you all might have seen me teaching math in the past but I'm promising you that we are going to take complete syllabus of maths on this channel for free okay happy so anyway let's quickly go on to the things so
• 02:30 - 03:00 first of all before I I know many of you know me but still I just want to introduce myself so my name is vumar and I'm basically from the mechanical team of PW gate and basically how you can or why you can actually trust me okay so basically I have the best ranking of A4 in Gate examination in the engineering Sciences paper where maths is relatively tough compared to the classical branches okay so we have this uh I have this gate all off four and previously I have worked with hpcl and also BRC Baba Atomic Research Center which I think
• 03:00 - 03:30 recently the current short list has come out for the interviews and you guys are preparing so I have been working with them for past some time then I have G qualified eight times in my career and of course 12 papers because since 2021 it's been double papers you can answer means basically you can write for two papers what is exsl I'll tell you okay engineering Sciences I'll just give you a hint just give me some time so I've been teaching teaching gate related things for past 7 plus years okay so before that I was teaching J Advanced actually but in the past 7 years I have moved towards gate examination so this 7
• 03:30 - 04:00 plus years is regarding my gate teaching experience actually okay and subjects that I teach generally is of course engineering maths which many of you might have seen then fluid mechanics basically I'm from the mechanical specialized subjects okay so I'm going to take here uh this fluid mechanics then basic app thermodynamics heat transfer these are kind of subjects that I generally deal with okay and as per current discussion I'm going to take engineering maths on this uh platform for now and it's going to be a complete course and then what all branch of
• 04:00 - 04:30 syllabus I'm going to cover that's the next question so I'm going to cover the syllabus of mechanical civil ECE triple computer science it XC and also data science okay and of course it's your responsibility to check which part of cabus is in your branch and to study only that Parts like for example in computer science you have only linear algebra probability and statistics and basic calculus okay it's not advantageous for you to study Vector calculus or complex calculus things like this okay so but since I have to C all
• 04:30 - 05:00 your needs I'll teach the complete syllabus but it's purely your responsibility to check what part of the syllabus is exactly matching with the content and go for only those set of videos okay so now anyway coming to the story now recently we have launched a telegram channel for exclusively English so whatever the things that's going to happen on this batch are going to be notified on this channel so all of you can just scan this QR code it will straight away take you that channel just click join you can join the gate English
• 05:00 - 05:30 Channel of course okay and this is my uh PW what maintain telegram group actually so if you click this you can join my group where already some 800 plus 860 plus students are there so definitely uh you can just directly get in contact with me and if you have any queries or something you can definitely post it in this group so this group is exclusively for gate Wala English Channel and this group is for V of course my group actually which is maintained by the students and me collaboratively okay so I'm your previous students thank you fine so look
• 05:30 - 06:00 here what is the syllabus so first of all I'd like to just give the sequence of the syllabus that I'm going to cover in this batch okay so that you all can have a quick follow on how things goes on so first initially I would like to start with basic calculus but because of slight request from students I have slightly changed the sequence first I'm going to cover linear algebra okay and when I am covering linear algebra please don't see matrices as a you know underrated
• 06:00 - 06:30 chapter actually okay maths is uh linear algebra is one of the very important chapters and in fact if you see there's no paper almost in Gate history that there's no question with uh that there's no question in linear algebra every paper you take definitely you you'll have some linear algebra questions from that okay and the kind of approach we follow okay even when of you many of you when you writing codings and all you try to type uh you try to work with arrays okay so when you are working with arrays how to write the elements and all I'll just give you some brief hint and definitely after finish of this course
• 06:30 - 07:00 you'll definitely find yourself in a better position in linear algebra okay and second I would like to go is basic calculus or I would like to call this as single variable calculus single variable calculus and third is the multivariable calculus of course we'll talk about functions with two variables three variables any in general TS is for n dimensional space we'll see all these things so once this calculus part is done I would have gone to probability and statistics also but I
• 07:00 - 07:30 don't want to cut the sequence so I'll teach differential equations both ordinary and partial okay differential equations and this is both ODS and PDS okay ordinary differential equations and partial differential equations as well okay because many times I've see people they skip this PDS but if you look at papers like uh you know basically if you look at papers like civil okay or mechanic in fact or
• 07:30 - 08:00 even in EC sometimes okay or XC exclusively there are a lot of questions from PDS okay so please don't again look back PDS PDS is very interesting stuff actually okay you have the waves okay waves vibrating vibrating string or diffusion some heat is Flowing or some if you let's say for example if you spray perfume how the perfume propagates we can see the diffusion equations and all in PDS and I'm telling you I'll connect all the subjects to the Practical things on your technical subject okay it's not like I just teach this integration is this this no it it works out but at the same time I'll try
• 08:00 - 08:30 to connect all the mathematical formula that we D here to the physical things okay like simple example you all know force equal to mass into acceleration then this is actually a differential equation correct because you have this DV by DT okay in fact if you deal in three dimensional space you have vectors then this is a vector differential equation so how to deal with all these things okay so we will connect then once differential equations is done we'll go for Vector calculus again very interesting thing and the advantage with vectors you can
• 08:30 - 09:00 solve in all the three directions simultaneously okay so this is very beautiful chapter and then after we'll look at complex calculus complex calculus and 7 you can see after finishing the complex calculus but it's roughly like the finishing of the calculus starting from the second to the sixth chapter then we will introduce probability and statistics which is the current days hot
• 09:00 - 09:30 cake of course probability and statistics so which I would like to put some more importance on that because ultimately every data science needs stats so once probability and statistics is done we come to the most powerful chapter numerical methods why I'm calling it more powerful I'll give you okay at introduction of each and every chapter we'll see the applications of that particular chapter and we'll have this numerical methods as one of the most powerful chapter of course okay then coming to the ninth I would like to
• 09:30 - 10:00 discuss a bit of transform Theory transform Theory which is a bit again important for all the e guys out there so in transform Theory we'll work with lapas and also for year space okay for year transforms are very much helpful to you so we will see how we calculate not only calculating physically we'll see what does a for transform can do physically okay and of course if time permits I'm not sure pretty much but if time permits I'll also do fast four year transforms okay
• 10:00 - 10:30 the ffts okay so this is badly the syllabus and of course four years is okay uh I would like to add one 10th chapter which would make the list sequence and say I would like to add it for you here so this are the topics that I'm going to cover so I would like to roughly divide it into 10 modules okay so that we have linear algebra single variables multivariables then differential equations both ODS and PDS Vector calculus complex calculus 7 8 9 and 10 of course so this is the syllabus that
• 10:30 - 11:00 I'm going to cover in this particular module okay are you all clear with what the syllabus so write total lecture needed also roughly 40 maybe okay yeah we'll cover full mathematics okay no issues but please type in the chat box is this clear to all of you are you all clear with what is the sequence that we are going to flow follow and what are topics we are going to cover in this particular lecture uh I mean in this particular module in this course
• 11:00 - 11:30 yes everyone so please type in the chat box I think all of you are busy in writing notes correct writing these topics names don't worry this notes will be available to you you have the links in the description so if you go there you can just get the link for the notes for the place where notes is posted okay no issue at all I want to know important chapter for civil engineering yeah I have done a video what silabus is there in which particular Branch so please refer that video on the same channel we have the video of course okay uh class notes in group I'm not sure but
• 11:30 - 12:00 we'll see okay but there's a place where all this is there so the link is in the description fine so now coming into the offences of course the class notes itself will be self-sufficient for gate examination and also all other technical interviews of course but still if you generate some interest during my teaching and if you want to refer some books definitely I would like to suggest a couple of books for you so one thing is BS G which is very well known to you higher engineering maths of course you have the latest edition 44th editions so this you can use for problem solving and
• 12:00 - 12:30 concept building Concepts and problem solving this is one standard textbook which you can follow and if you want to connect things practically to your gate to your technical subjects then this book Advanced engineering maths by Ain kig is fantastic book phenomenal book actually so this book is for practical connectivity practical connectivity Okay so so this is the book
• 12:30 - 13:00 for the Practical connectivity like let's say for example if I learn gos elimination method why you needs elimination method in your life this book answers you that question okay so when we have when we develop linear systems why gos elimination is one powerful technique or what are the pitfalls in this gos elimination method so all these things we of course uh get in this Cas so for tily complex variables yes you have you do have complex variables see this batch is for all branches combined together and I'm telling you you please look at your syllabus and if only to those chapters
• 13:00 - 13:30 during the courses okay and this of course linear algebra basic calculus I mean the single varable calculus probability and statistics is there for every Branch okay now before I start the classes I'd like to give you information on one important stuff actually this Unity okay so w u n t y okay so what is this Unity what why PW has actually developed it I'll just give you an information so normally during a class you guys actually keep lot of comments okay and at times perers even they respond or they'll tell the answers and
• 13:30 - 14:00 you guys will again forget that after some time okay so Unity app will actually keep a record of all your chart and all your uh doubts actually okay like let's say for example if I'm teaching something if you have a doubt then you can actually type the doubt in a particular way okay so you can see here I hope you can see when you have a doubt in this chat box if you if you want to ask me some doubt before the actual doubt you can just type this exclamation and doubt and then start typing your doubt like let's say for example if you want to ask you what is a
• 14:00 - 14:30 matx okay then exclamation doubt then you can ask what is a matx the advantage of asking this is this creates a track of all your doubts like normally if you click on this Unity app it asks you for sign in okay login with Google account you can see there and if you actually log in okay I have logged in with mine this is how the page comes okay now during live classes we keep some polls you guys will participate in the polls obviously so complete your poll history over the complete year not just this class a complete period gets stored in
• 14:30 - 15:00 this poll accuracy okay and you can see here polls attempted classes attended how many classes you have attended throughout the year and let's see if I have some doubts you can just go forward then you can uh see basically what are the what are doubts you have what are the questions you have asked all this track record is maintained in this Unity uh website so please for this uh please go to the link Yu I sorry Yu nit y unity and then you can just log in with the sequence what I have taught you it would definitely help you out okay so the only
• 15:00 - 15:30 thing you have to do is when you are asking a doubt just before typing the doubt this exclamation doubt similar if you want to make some notes you can write exclamation note and then make some points those points whatever you do during the class will get stored in this Unity app even if you lose your class notes you have a backup of the things that you have okay so these are the things that I would like to proceed now one thing that I would like to tell you before I start the lecture is if in my class whenever you have any doubt any
• 15:30 - 16:00 okay you can always ask me okay let's say for example don't feel that it's a small doubt if I ask what he will think no nothing okay you are here to learn we all are here to learn in fact so if you have any doubt please ask it immediately I'll be I'll be the happiest one to answer okay never think of anything just ask it okay during the live class you can interrup me no issue and so with this introduction I'm going to start the first lecture formally of this particular course this is the introduction of this course these are the reference textbooks that you can
• 16:00 - 16:30 actually follow and this is the complete syllabus that I'm going to capture in this case okay and the branches which are the syllabus of all branches what I'm covering is mechanical civil ECE Tri computer science and uh this XC and data science of course these are the things clear so will math when math syllabus gets over start I mean sorry please let me start first okay fine schedule is already uploaded in the app uh in the free batches app in the links if you you just look at the links you can see detailly
• 16:30 - 17:00 clear fine so shall we start the first lecture are you all okay so as I told you the first topic that I'm going to start is linear algebra and this is lecture number one so are you all ready to start this lecture so please type in the chat box yes so please type in the chat box fast every one of you should type in the chat box are you all ready to start this course
• 17:00 - 17:30 yes or no fast ready complete math or just few topics of math complete math okay yes okay fine so the topics that I'm going to cover today is matx and its types as I told you I won't expect any prerequisite from you we'll start from zero okay so we'll start from the definition of a matx and we will go in uh advanced stuff and then we'll also see a bit of Max algebra with time permits today so first of all before
• 17:30 - 18:00 starting this matx and all why do we need matrices okay why it's advantages for us or what difference matri make in our life can anyone tell me can anyone tell me why you are studying linear algebra because two days back vesa has made a video that linear algebra gives you two to three marks in Gate exam so we have to study linear algebra correct no what difference mates make in your life anyone I'll ask you some questions okay you should answer it's okay if you don't
• 18:00 - 18:30 answer it's fine but try to answer them why you learning matrices to solve linear equations good so that we can calculate the rank okay maybe after calculating rank you want to do something mat are useful in solving vectors good yeah we do vectors for marks on job to calculate the rank okay so basically I'll just tell you one thing first as many of you have told solving linear set of equations
• 18:30 - 19:00 solving linear set of equations and if you make a word if you see I'm just writing linear set of equation I did not write the word algebraic because you can solve even differential equations sometimes okay so that's why because there are plenty Concepts like quad quadratic form diagonalization all these things they keep you lot of stuff actually in solving linear set of equations actually okay now not only this we have lot of applications in medical imaging
• 19:00 - 19:30 the reason is actually if you take your human body I'll just tell you one small interesting thing this body is symic okay you know apart from just few organs the body generally looks symic okay so let's say if you generate a point here you can reflect this point here about a particular axis okay so data science nowadays you deal with lot of partition matrices all these things so data science linear algebra techniques are very powerful actually so if I keep talking almost each and everything deals with matrices okay so linear algebra is
• 19:30 - 20:00 very important thing to learn in day-to-day life of course and of course solving linear equations I just give you some idea if I give you a equations like 2x + 3 y = 5 and maybe 6X + 8 Y is equal to 14 something like this if I give you you can solve very easily so because these the two equations in two unknowns okay two equations with two unknowns correct but what if if I give you some 100 equations with 100
• 20:00 - 20:30 unknowns 100 equations with 100 unknowns then ultimately your life becomes complicated right so to eliminate all these things we use linear equations of course and now coming to the definition of a matx matx an array of elements an array of elements you know basically we work with arrays in nowadays you know C++ python Java whatever we will work a lot with arrays
• 20:30 - 21:00 so arrangements so an array of elements in horizontal lines in horizontal lines called row or vertical lines in bracket they're called as columns is called a matx is called a matx and in general matrices are denoted with capital
• 21:00 - 21:30 letters of course so they are denoted with capital letters in general they are denoted with denoted by capital letters like for example you see I would like to give some classical examples which you see a is equal to okay some 2 3 4 6 whatever so here we have some elements 2 3 4 6 we have arranged them in row in lines which are basically rows
• 21:30 - 22:00 and also some vertical lines which are columns of course okay similarly the ENT inside the Matrix can be alphabets like for example V I'm using my name because I'm teaching if you teach you can use your name loves don't worry much I write maths here okay so I know you guys may be expecting some other name but obviously I love maths okay so that's why I've been teaching it and then you can have lot of things like this maybe uh 1 2 3
• 22:00 - 22:30 you can have ET end of my thinking capacity that's it okay so these are the matrices of course we keep generating elements in rows and columns and there are lot of types of matrices of course okay so we will see each and each type and what is the speciality of each type we will also observe okay now let me tell you one thing this is one form of representing a matx but unfortunately let's say if you have 10,000 elements in a matx okay so let's say this is 3x3 so I'm happy to write each and every element but let's say if I have a 100 by
• 22:30 - 23:00 100 elements okay or 10,000 elements in a matx then it's not possible for me always to write all the 10,000 elements okay so mates are also defined in another form okay so sorry so also teach us in English medium uh not as of now but first let us finish the English okay then let's see order or size of a matx looking at these definitions please don't feel I'll teach only these definitions we'll go to Advanced things but I want to start from
• 23:00 - 23:30 zero Okay order or size of a matrix so ultimately if you take a matrix as the name tells you size in every Matrix there are some finite number of horizontal elements and some finite number of vertical rows also okay so this number of horizontal rows and vertical lows decide the total number of operations on that Matrix like let's say if you're adding two matrices okay then whether you can add or not if you're adding how many additions you have to do these kind of or if you if you have two matrices if you're multiplying those two matrices what should be there okay so these things the feasibility of the
• 23:30 - 24:00 algebra of the matrices depends upon the size of the Matrix actually clear so order if if a matx is set to be of size set to be of size or order M by n it's read as M by n of course you can write this normally this into sign is read as m M by n
• 24:00 - 24:30 if it has M rows by this time you know what is a row and Y Columns of course okay and it has y Columns of course here so like for example if I say a is equal to 1 2 3 4 5 6 for example if I write this Matrix then ultimately you all can see this Matrix has two horizontal l two horizontal lines of course this and this so the first number always denotes the
• 24:30 - 25:00 rows and if you have checked the vertical lines you have three columns so this is 2 by three of course okay so is it useful for diploma students yeah definitely Okay because I'm going to start from zero okay so let's see this is 2x3 so if this is 2x 3 it tells you that this Matrix has two horizontal lines and three vertical lines of course and total number of elements in this is 2 into three which is six of course here okay so in general a
• 25:00 - 25:30 matx of size M by N is a matx a let me write here a matx capital A a matrix capital A of size m byn is denoted as a m by n of course okay so it tells you that a has m o and also n Columns of course okay sorry sorry if I I believe
• 25:30 - 26:00 if you're teaching English then I'll be more useful it's okay no issue okay so once this batch gets finished we'll plan something similar okay now let's see as I told you now when the Matrix size is simple like 2 by three then totally you have six elements inside the Matrix you'll be able to write each and every element but let's say if the matx size is 100 by 100 or th by th sometimes okay if you're doing finite element methods okay how many of you know finite element methods anyone of you from mechanical or civil fem the popular fem finite element methods or cfd competition fluid
• 26:00 - 26:30 dynamics do anyone of you heard these names no heard but don't want toil right because again I'll start asking some questions no okay no issue if you don't know it's fine so basically there come situations in life okay okay no issue if you don't know it's fine so there come situations in life where you can you may have to solve some 9,000 by 9,000 order matx okay where the total number of elements are 81 into 10^ 6 okay huge we need super computers or very high
• 26:30 - 27:00 speeded Computing and all to do that okay so but anyway my main intention is whenever these numbers are large then ultimately the number of elements in this are also very large okay okay no issues okay so when this number of elements are large how do we denote this numbers that's what okay like example I'd like just give you an example a is equal to let's say if I write this Matrix 2 3 4 5 6 okay then 3 4 5 6 7 4 5 6 7 8 and maybe
• 27:00 - 27:30 uh say you know uh this is 5 6 7 8 9 let's say I'm stopping here maybe it's of size how much 1 2 3 4 5 so 4 by 5 Matrix and you know it consists of 20 elements okay so whenever it consist of 20 elements let's say if I don't want to write this matx when I'm giving the question or let's say in Gate examination if I'm giving equation I don't want to mention this complete Matrix I can also mention this
• 27:30 - 28:00 matx in another form look this same matx can be represented as a equal to a i j where a i j is equal to I + J for all 1 less than or equal to I less than or equal 4 1 less than or equal to J less than or equal to 5 if if I don't worry I'll explain you each and every step
• 28:00 - 28:30 don't worry now if I write this line or if I write this matx both are same actually okay first thing whenever I write this element a i j look a i j this first subscript which is here I this denotes row number this denotes R number and whenever you have this J this J
• 28:30 - 29:00 denotes J denotes column number okay now let's see if I ask you what is the third or second column element for example randomly okay when I say third or second column element you guys need to calculate third because I told you first subscript is the r number second column element okay so it's a second column now I have given here a i j is equal to I plus okay means any element in the I jth column is equal
• 29:00 - 29:30 to the sum of row number plus column number that's how I have defined my matx okay this definition is given by me so if I want to calculate a 32 the element in the third or second column is nothing but 3 + 2 which is equal to 5 now you know how to identify third or second column come here third o is this basically maybe I'll uh use different color so this is the third this is the second column the guy at the intersection is this is the third or
• 29:30 - 30:00 second column element the value is five and here this 3 + 2 is also given as five okay so like that you can see I changes from 1 to 4 that means I can take the value I means not me this I okay this I can take the values from 1 to four that means I value can be 1 I can be 2 I can be 3 I can be four actually okay that tells you there are maximum four rows in this Matrix so 1 2 3 4 okay okay and J J is going from 1 to
• 30:00 - 30:30 5 that means this Matrix has got five columns 1 2 3 4 5 and any element you can match definitely the value given by this expression and the value given by this uh in the matx will be equal actually okay like for example if I want to calculate fourth or First Column element okay so if I put I equal 4 and Jal 1 a 41 fourth or First Column element is equal to five and you can see this is five of course here okay so did you all understand how we can actually represent mat without writing the full matrix by using
• 30:30 - 31:00 this in general if I want to specify this a i j will be given as in general in general this a i j will be some function of I comma it can be I + j i power J whatever if I want to set the Matrix I can explain whatever I want but I'm just telling you this matx any element in the Matrix can be given as some function of I comma J okay now if I want to generate a th000 by th000 matx for example let's
• 31:00 - 31:30 say every element is connected in a particular way then I can say this changes from 1 to th000 this changes from 1 to th000 a is equal to that function what I have generated I can give it that's it clear did you understand without writing the complete Matrix just by writing one line can you identify and the reason why I'm teaching this is many of the gate questions come in this format okay they are not interested to give you the full matx they'll give you this and with this you can do what what all you want we'll see in types of matrices and all clear to all of
• 31:30 - 32:00 you everyone so please type in the chat box is this clear did you all understand this notation of writing the matx where you define the row and column numbers and then you define the element at random the general element okay 100% clear some love symbols and 100 symbols are going on good okay guys I'm telling you at any point
• 32:00 - 32:30 if you have any doubt please ask me I'll be happy to explain okay please don't think it's a small doubt or big doubt or doubt doubt is a doubt that's it okay then let's see let's move on to the next case so this is how we generally go for an example now shall I give you some funny exercise fun activity we'll do okay so I just want you to figure out one thing the sum of all the elements
• 32:30 - 33:00 the sum of all the elements of a matx the sum of all the elements of a matx y where a equal to set of a i j such that a i j is equal to i+ J for all let's take a square Matrix for Simplicity or maybe if you want I can complicate the question less than or
• 33:00 - 33:30 equal to 4 maybe four J less than or equal to 6 let's I'm taking six here okay because if I give five you will ultimately add these numbers and tell me the answer but I just want you to calculate this okay is Dash let's say if I'm giving you this question for example this is actually one of the G questions not exactly these numbers they have given some different numbers but this function is same I plus J I'm connecting a poll integer type ultimately answer will be integer because you can add integers I'm giving you 150 seconds of no 120 seconds of
• 33:30 - 34:00 time I'm kind enough so starting the poll you can enter in the chart box yes do it now you know how to form the matx okay so if you form the matx and if you add the sum of all the elements what you're going to get that's the question it's actually a prvious gate question okay I'm not teaching anything for just to tell you that I know maths everything that I'm teaching you is in the gate examination okay come on quick be fast okay meanwhile I'll write
• 34:00 - 34:30 matx painfully look now you have to add all these numbers and tell me what is the answer that's it okay all numbers of the
• 34:30 - 35:00 matx the summation of all the numbers of the matx actually come on quick think time is running up fast
• 35:00 - 35:30 114 maybe I don't know so can we write from one no because the first to First Column element is 1 + 1 because first to First Column a 1 1 so 1 + 1 2 okay so the first element is two matx WR in function form F of I comma J yes of course time is up time is up come on fast fast fast near 114 maybe I don't know I have
• 35:30 - 36:00 to check okay good so let's see one thing first of all if people might have been little smart many of you might have answered this question 134 okay look let's see let's see it's okay 144 good next 124 come on I think you guys are watching IPL A Lot H fixing the targets 114 134 144 good CH let's see here I'll tell you
• 36:00 - 36:30 one thing guys look if you add these numbers okay let's say I'm adding these numbers how much do I get of course I'll tell you a nice way to solve this question but just I'm giving some idea 2 + 3 5 5 + 4 9 9 + 5 14 14 + 6 20 20 + 7 27 this is 27 now if You observe little carefully If You observe little carefully this number is one more than this this number is one More Than This similarly every number is one more than its previous s
• 36:30 - 37:00 number okay if this is 27 can anyone tell me what is this sum of these values every number is one more than the previous number so totally you have six numbers so can anyone tell me what comes here actually this kind of questions are coming in aptitude okay last day also I mean 2024 also they have given you one question there are two sets of roads on one side of the road you have numbers house numbers like this on other side you have house numbers like this what is the difference and all okay if this is 27 what is sum of these
• 37:00 - 37:30 numbers 33 36 24 I see here if you adding these numbers you're getting 27 every number in this is more than its previous by one okay so ultimately it should be 33 then following the same logic these numbers are one more than the previous number so this is 39 and following the same logic again these numbers are one more than its previous numbers so this is 45 and if you add all these things you would get 4 2 4 7 10 14 144 that's
• 37:30 - 38:00 it okay yeah Vista you are correct kusaga I think you missed some 10 here so this is what you're getting right 144 now tell me how much time it takes to solve this question forget about matrices you you guys don't need not know matrices to solve this tell me how much time huh silly errors these silly mistakes are very very costly in exam you know how much costly they are I missed two times all India rank one
• 38:00 - 38:30 because of those silly mistakes okay so anyway I'm happy with what I have so this 144 is the correct answer now tell me 120 seconds are enough to solve this or not enough to solve this he fast here please tell me these 120 seconds are good enough to solve this question or not you might have done lot of additions and subtractions for past 10 15 years of your life correct yes enough then why you all didn't answer
• 38:30 - 39:00 because I've given some poll time constraint okay so this pleasure was there in exam also the same thing happens but don't worry just trust your Basics you can do it okay so but anyway this is a small question so I have done it this way but let's say instead of this if I give this numbers as 2,000 and 1,000 things like that I cannot keep doing this adding business okay then what I have to do okay so anyway the answer is 144 so let's check it up what we can do in general let's say a is equal to set
• 39:00 - 39:30 of a i j m by n let's say for example you have M by n a i j is equal to I + J for all 1 less than or equal to I less than or equal to m 1 less than or equal to J less than or equal to n okay so if you just look at this Matrix let's see I'm constructing this Matrix all of you will see first of First Column element will be 1 + 1 actually okay will you all agree because first First Column if you
• 39:30 - 40:00 put IAL 1 and Jal 1 it is 1 + 1 it gives me two first second column element first to third column element this goes on first to nth column element okay I have this then coming to the next stage second o First Column element second o second column element second R third column element this third First Column third second third third column plus n and so on let's say this
• 40:00 - 40:30 goes on you have M rows in this matx so m + 2 m + 3 and so on m + n actually here so let's say this is the matx now you want to calculate the sum of all these numbers of the matx for example let's say Okay then if I add first to what all I'll get let's check it up if I add all the elements in the first to look in the first term I have one in the second term also I have one in the third term also I
• 40:30 - 41:00 have one so look like that I have total n columns so total I have n times of one okay so all the ones in the first place they add up for n times of one because there's one here one one one and so on next coming to the story of the second term here you have one here you have two here you have three so so on till you have y so the sum of all the second terms is nothing but sum of first y natural numbers yes or no will again means it's nothing but 1 + 2+ 3 and so
• 41:00 - 41:30 on plus n of course will you all agree this yes or no please type in the chat box why I'm telling this is if you are trying to plot or if you trying to code something in C or C++ for example or python okay then we should have certain understanding in the indices because many of you will commit mistakes with the for loops and you know basically uh F Loops or if Loops okay in general if if is simple of course you'll just check the condition but when you're are doing F Loop GRS many times people write the mistake of indexes indexes okay they'll
• 41:30 - 42:00 start from I equal to one they don't close at n they do these kind of mistakes very frequently so I'm just telling you this will you all agree with this please type in the chat box understood 1 * of n plus sum of first 10 natural numbers that's what you have okay clear fine then coming to the second term second here you have two here you have two again two and twos so total n *
• 42:00 - 42:30 of two is what you have plus again coming to the second term again story is same 1 2 3 and so on you'll have n so sum of N natural numbers if you keep doing this business for the next to you'll get n * of the plus 1 + 2 + 3 and so on plus n actually here okay now if I just extend it to my understanding the last to gives me npe of this M plus because there are m n actually
• 42:30 - 43:00 okay oh sorry n number of M's plus 1+ 2 3 and so on N of course here now sum of all these elements is nothing but sum of all these elements okay now let's see sum is equal to sum of all elements in general this is denoted as summation I = 1 to M summation J = 1 to
• 43:00 - 43:30 n a i j this is what you'll denote actually okay you'll see of course right now just take this I'll explain you how this comes okay first you'll keep Jal to 1 2 3 and then again keep I equal to 1 2 3 we'll see how these things happen but this value is equal to now if I add all these first terms I can take n common from each and every term okay so like for example n of 1 plus n of 2 plus n of 3 plus and so on n of M plus now this
• 43:30 - 44:00 term is common in each and everything and total you have M number of rows M number of rows so M * of 1 + 2 + 3 and so on plus n this is what you will get okay and if you simplify this ultimately you can see n if you take common this is 1 + 2 plus 3 and so on plus M this is some of first M natural numbers so n if I take common I have M into m + 1 by 2 plus M into then this is sum of first n natural numbers n + 1 by 2 if I take MN
• 44:00 - 44:30 by 2 Common from both the terms I have m + n + 2 of course okay so that's the expression for the sum of the numbers clear so therefore sum is equal to m n by 2 * of m + n + two of course clear so this is what you have got it did you understand okay this I have generated
• 44:30 - 45:00 only for this function I plus J if I if I have different function then definitely elements would have been changed okay did you understand how we are going to generalize the things yeah so now using this formula shall we check the previous result shall we check whether this is getting correct or not because the function is same I plus is only what I have given okay look I'm telling you we have d a formula for the general matx so
• 45:00 - 45:30 let's examine since this element is I + J and this element is also I plus J now that formula what we have DED should hold for this okay now let's see for M = 4 n = 6 because it's a 4x6 Matrix sum should be equal to m n by 2 into m + n by 2 okay so 4 6 are 24 by 2 into 12 so 6 144 which is this
• 45:30 - 46:00 actually yeah I got it okay you didn't understand fine no issues look basically uh till here did you understand writing the matx did you understand let's go in a sequence okay if you don't understand something we'll go in a sequence and we'll start at that point where you got doubt so let's see this is what you have is this clear till here first of all is this clear till this point writing the matx first of
• 46:00 - 46:30 all quickly please type a bit fast okay yeah now if this is clear is this clear to all of you till this point is it clear till this point is it clear to all of you till this point is it clear to each and everyone yes yes yes okay then let's see I understand
• 46:30 - 47:00 F about the summation for some time let's say I'm adding all these things if I add all these things let's say this is the first these are the first terms and these are the second terms if I keep adding all the first terms I can take n common from all the first terms okay so if I take n common I have 1 + 2+ 3 and so on plus M okay so if I take n common here I'll have 1 + 2 + 3 + 4 and so on plus M so n if I take
• 47:00 - 47:30 common I'll have this expression and sum of first M natural numbers is M into m + 1 by 2 Okay so this n into sum of first M natural numbers M into m + 1 by2 plus now second we have these terms this term is common in all the steps but total how many O's you have it's a m byn Matrix so total you'll have M times of this term okay so M * of this this term which you have here so sum of first 10 natural numbers is n into n + 1 by 2 so MN by2
• 47:30 - 48:00 is common in both the terms if you take m + 1 + n + 1 m + n + 1 + 1 is 2 that's what clear yes or no please type in the chat box okay now it's clear everyone Sam Angel Hamma Yami sorry study tips for
• 48:00 - 48:30 mechanical students maybe mechanical student home okay hi Krishna everyone got it clear to everyone who all has doubts now it's clear good so we have expressed the same expression but because of logic we have told the answer is 144 which matches actually okay now let's see one more
• 48:30 - 49:00 thing summation part okay some of you have asked please explain this summation I'll explain okay look here look let's say I have this Matrix a is equal to a11 a12 A2 1 a22 let's say for example I'm explaining with 2 by2 so that it is easy so sum is equal to you all know a11 + A2 1 + a12 + a22 this is what you have okay a11 + A2 1 plus or maybe a11 + a12 A2 1+ a22 whatever then
• 49:00 - 49:30 if I write this notation Sigma I = 1 to2 = 1 2 2 Sigma J = to 122 a i j if I write this let's see what all we get okay then it has two summations one is for I and one is for J keep the I summation as it is and let us expand the J summation okay J is going from 1 to 2 what does that mean a i 1 + a i 2 yes or no will
• 49:30 - 50:00 you like because I'm talking about J expansion so if I keep I fixed then at a particular J if I expand this expansion I would have a i1 because J is first replaced with one this is summation so plus then I is going from J is going from 1 to two so after one it J will take the value of two so this is what you'll have okay now if you expand this summation so first I equal to 1 this gives me a11 +
• 50:00 - 50:30 a12 plus again I can go from 1 to 2 also so therefore if you replace I with 2 actually we have a21 plus a22 now you can see this expression and this expression both are same clear so inverse of matx 33 matx you want short te okay fine wait till we go for inverse okay I'll definit it will teach you okay so see here did you
• 50:30 - 51:00 understand how the summation Works actually how this notation Works actually everyone so please type in the chat box did you all understand how this notation works okay so first it runs on the inner summation and then it takes up the outer summation that's it okay yes did you all get the idea now
• 51:00 - 51:30 how to do so writing time please I TK uh you can write running notes and anyway these notes will be uploaded so no issues fine I'll give you one logic here here you asked me to teach short takes right I can teach I know many of course you can see as this course goes on I'll teach you many tixs but in general I just want to tell you uh one important thing gate exam is not about shortcuts okay honestly fact okay because I have
• 51:30 - 52:00 myself have secured all India four I know what is gate examination see I'll tell you even funnily I see many thumbnails circulating on YouTube uh this question th seconds I'll tell in English so solve this question in 10 seconds solve this question in 3 seconds 2 seconds 1 second and people say just by looking also you can say of course we do say but I'll tell you one thing honestly let's say your gate examination is set for 65 questions okay and this 65 questions uh should be solved in 180
• 52:00 - 52:30 minutes okay so roughly if you calculate you have 2.7 or 2.8 minutes for each question on an average okay so let's say if you learn so many shortcuts and if each question can be solved in 10 seconds of let's say 10 seconds calculation is easy so 65 questions should be solved in 650 seconds which means within 11 minutes your gate exam would have been done yes or no correct will you all agree with me if I keep teaching shortcuts I'm just telling I'm not criticizing or I'm not against anyone but I'm just giving you the basic idea if I have shortcut for solving each
• 52:30 - 53:00 question then my gate exam would be done or maximum in half an hour my gate exam would have been solved all 65 questions yes or no correct forget about a 1 we'll see what happens okay but why I is give you 3 hours to solve this gate paper the reason is shortcuts of course if you
• 53:00 - 53:30 very very familiar with few Concepts you can apply shortcuts for those questions but in general gate is not an exam of shortcuts Okay the reason I'll tell you if you go for a single subject you can maybe remember some shortcuts and you can do well with the shortcuts okay but let's say in math itself let's say in linear algebra I have taught you five shortcuts let's say for example okay in linear algebra I have taught you five shortcuts now there are total 10 chapters you can see in the beginning I have given the list so this 10 chapters into five in math itself you'll have 50
• 53:30 - 54:00 shortcuts okay and totally let's say you have some 10 subjects total you'll have 500 shortcuts is it possible for any human physically to remember all the 500 shortcuts and the conditions when these shortcuts can be applied exactly and then apply in the exam can anyone do this can anyone remember all these 500 shortcuts with the conditions when they can be applied and can they apply in the exam correctly is it possible of course I do teach okay no
• 54:00 - 54:30 issues but I never ask or compulse my students if you don't know this shortcut you will not get A1 no no need okay I'm happy with A4 and I'm doing well in my life but I don't remember many shortcuts okay so I'm telling you now because see please tell us something interesting incidents doing your get preparation Paka yeah I'll tell Will GC English batch come yeah it's the plan not possible okay so uh I'll tell you
• 54:30 - 55:00 some nice things in between the classes I don't make it boring because I can tell you it's already 55 minutes since we started class it's almost 1 hour you can check the timer okay we started sharp at 7: it's almost 7:55 did anyone of you feel bored at any point in this no right see I'm telling you now gate is not about shortcuts to be frank okay and of course lot of thumbnails and all the C everywhere but please don't believe in all the stuff the reason is the problem with when people say this is see let's
• 55:00 - 55:30 say for example if I chilled for 2 years in my life I secured good gate rank after securing good gate rank if I say that I used to study 12 hours a day people believe me okay because I got rank unfortunately okay so people believe this I'm telling now like if you see whatever the strategy a particular topper tells is valid only for that particular person okay because the way people study things people understand things is different no
• 55:30 - 56:00 two individuals may understand the same thing in a same way okay so I'm telling you people actually get people have different ways to learn okay and people follow different strategies but unfortunately when Toppers say that this is the strategy I have followed people instead of learning in their own style they forget their style and start applicating some other style okay which actually delays your success clear so I'm telling you please if you find if you can figure out this is the way that I understand things you please do it your way you'll get good things in life
• 56:00 - 56:30 okay and also I'll give you some nice useful things in maths as we progress okay so lot of mates I have taught I just want to ask you some question you all know what does this give you what does this equation give you what does that equation give you what does that equation give you
• 56:30 - 57:00 line okay so this is a straight line good straight line nice many of you have answered straight line what does this equation give you what does this equation give you Circle correct it gives you a circle good can anyone tell me what does
• 57:00 - 57:30 this equation gives you can anyone tell me what does that equation give you what does this equation gives you any idea no idea circle circle for previous thing
• 57:30 - 58:00 or this nothing it gives something no why nothing it gives something maybe we don't know what it gives no it almost gives you a square look I'll just show you normally you can use this normally I keep uh teaching people in my classes so Desmos graphing you'll say Desmos is basically a graphing calculator where
• 58:00 - 58:30 you can plot lot of curves so I'll show you maybe x² + y Square equal 1 I hope you can see the screen okay it's not blurred okay forbidden 404 good kids gaming see some kids are much interested in playing games huh x² + y squ is equal to 1 okay so you know there's a small circle which got generated here clear so this is a circle and you know ultimately this point is one and this point is also
• 58:30 - 59:00 0a 1 and 0 comma minus one of course so let's check it up whether what I told you is correct or wrong X power 2024 plus y power 2024 is equal to 1 you can see what did you get can you see see this equation is plotted in a red color so in red color you have a circle this equation is plotted in blue color you can see
• 59:00 - 59:30 there's a blue it's a circle it's a square which circumscribes this circle actually did you understand it's a square can you see can you can you all see it's a square yes or no how sir you think okay how this can give you a square that's your job now think
• 59:30 - 60:00 okay I'll just tell you one thing circle with a square or only Square no no this is only a circle if I delete this equation that red color thing will go you can see okay there's no red color only blue color okay if power will be 2025 still a square anything beyond 202 not only 20 24 as long as this uh number crosses Beyond 100 or 200 it gives a square do you want to know I'll just maybe I'm also not very old let's
• 60:00 - 60:30 see do you want to know what does this SC gives you do anyone know what does this SC generates do you know what that curve generates anyone because I've been teaching matri
• 60:30 - 61:00 for past one hour so a bit of refreshment I have a lot of stuff like this I'm promising we'll give you one each day okay just one in between two USS somewhere I'll give you one Circle why because this equation is square now you're going back I'll give you a hint as of now you all are seeing the shape of this graph everybody now doing Desmos Parabola no
• 61:00 - 61:30 sagik no sir we don't know sir I'm giving a hint as of now you all are seeing this graph which is the shape which is generated by this equation as of now you all seeing the shape which is generated by this equation shall I tell you I'm very poor artist so forgive me for some bad
• 61:30 - 62:00 drawings this is what it gives and which you able to see exactly on the right side of your chat box shall we see whether we'll get that or not desos x square the usual stuff I'm deleting this okay I hope by this time you understood it's a square plus y minus square root of mod
• 62:00 - 62:30 x power 2 is equal to 1 you can see okay so that's the equation it's plotted in given color I have deleted all the curves to make you understand so this is a perfect cardioid okay in polar coordinates if you WR I is equal to a * of 1 - cos thet okay so all this important stuff nice stuff you can enjoy math is really
• 62:30 - 63:00 beautiful okay I'm sure if you are hating maths that means you didn't understand math in a correct way pakka okay same holds for physics chemistry any subject in general fine did you all enjoy this you can see this is the mathematical equation of this clear so next time when you go for people or person from mathematics Department you write I bracket x² + y- Ro of mod x square is equal to 1 close the bracket and you write U okay then people
• 63:00 - 63:30 understand what you're telling clear good so let's get back into the mat stuff again okay so anyway daily we'll spend some 5 to Six Minutes on these things so anyway let's see uh now we have talked about this then principal diagonal of a matx which is very much important actually okay the concept of diagonalization or
• 63:30 - 64:00 talking about some special types of matrices everything comes in this okay so principal diagonal sorry diagonal diagonal what is the spelling of diagonal diagonal okay principal diagonal of aax principal diagonal of a matx uh sorry everything fine with the graphs okay so let's get get back into the maths
• 64:00 - 64:30 then the line of elements a i j the line that passes through the elements or else you can say that passes through passes through the elements elements a i j where I is equal to J like for example
• 64:30 - 65:00 you take a 3X3 Matrix to understand the job easily you have the elements a11 a12 a13 a21 a22 a23 a31 a32 a33 of course okay so let's say you have this elements now if You observe one thing very clearly all these elements have the same value of I and J correct like for example you take I
• 65:00 - 65:30 equal to 1 and J equal to 1 that's the element in the first to First Column this element also has same values of I and this elements also have same value of I then this diagonal physically they won't be diagonal but this is a hypothetical thing so this diagonal is called the principal diagonal of the Matrix principle diagonal of the matx principal diagonal of the matx
• 65:30 - 66:00 now if I is less than J if I is less than J then a i j is called upper diagonal element uper diagonal element like if you see this example for examp okay now this is the diagonal and every element which which is above this diagonal will definitely have an I value Which is less than J which means I I denotes the row number
• 66:00 - 66:30 and J denotes the column number so definitely I less than J tells you that the element is sitting on the top of this upper diagonal you can check any of these elements definitely the row number of that element would be less than the column number and similarly if I is greater than J they come to the lower diagonal elements okay so if I greater than j a i j is called Lower diagonal diagonal element lower diagonal
• 66:30 - 67:00 element and let's see sum of all the principal diagonal elements principal diagonal elements is called is called one very important word this is very very important especially when
• 67:00 - 67:30 we go for IG values trace of the Matrix trace of the Matrix summation I = 1 to n a i i is equal to j i i is equal to trace of a normally in later stage we'll see something called the initial notations in such case you don't write the sigma this is called the Einstein summation convention but right now we don't focus on that as
• 67:30 - 68:00 far as our class standards this gate syllabus is concerned we'll put the summation Sigma I = to 1 to n is nothing but sum of all the principal diagonal elements correct because first if I keep a equal to 1 I mean I equal to 1 A1 1 then a22 plus and so on plus a NN is called trace of a of course okay so this called trace of a and the beauty with this St is let's say if you're transforming a matrix means let's say if
• 68:00 - 68:30 for example you're dealing with vectors okay and if you rotate the vectors then if you change the system the the matx a will take a new form but even in such case T will not change okay so we will see all these things when we go for I values okay so this is tace of Y of course now next definition which I would like to write is transpose of a matrix which is very much important again transpose of
• 68:30 - 69:00 aax okay so transpose of aax do you all know what is transpose of thematics can anyone tell me Abdul kotla Riki VHA yesu yaa y sorry har kishan Mr R sagnika Sam anoj anyone of you interchange corresponding elements about p diagonal do you mean about principal diagonal what is transpose can anyone
• 69:00 - 69:30 tell me changing rows and column interchanging can anyone tell me why this interchanging is important why you have to talk about transpose why interchanging is important can anyone tell me interchanging R and colums fine good enough yeah of course we interchange rows with columns and columns with rows of course okay so but can anyone tell me why this transpose business is
• 69:30 - 70:00 important yeah correct becomes columns columns becomes S I do agree rotate allows by principal axis what is principal axis you mean diagonal no of course maybe it looks the same but it need not be the same many times okay transpose is required to calculate the inverse of a matrix no without I don't want transpose okay without doing transpose I can still calculate inverse of a matx I don't need transpose to calculate inverse of the Matrix you all might have
• 70:00 - 70:30 done one thing in your bch or maybe in your plus 12 you have if you want to calculate inverse of a matrix let's say a b c d e f g h i you will take identity Matrix here by doing some operations finally after the operations you'll see left hand side becomes identity right hand side becomes exactly the inverse of this Matrix you might have done this then we never talk about transpose you remember you have done
• 70:30 - 71:00 this you write this Matrix then you write the identity Matrix here do some row operations or column operations whatever and finally get the identity Matrix here and then you'll get the inverse of this Matrix on this side you might have done this correct anyway we'll see later so let's see transpose why transpose is important is because few matrices they expose obviously in few substances if you look at materials and all in the laes OR basically in the chemistry organic chemistry or inorganics we'll see there are something
• 71:00 - 71:30 called stomas okay so they exhibit certain symmetries not only that there are plenty of applications of course so transposing means basically shifting of the things is important okay transposing means you know it's like transforming The Matrix into a particular form so we will see why this is actually important transpose of a matx so let's see first of all we'll see a matx b n by
• 71:30 - 72:00 m is set to be transpose of a matrix a m by n is set to be a matx a m byn if you you can better write this a i a MN is equal to a i
• 72:00 - 72:30 j m by n this is this is equal to bji n by m is set to be transpose of a matrix this if a i j is equal to BJ I for all 1 less than or equal to I less than or equal to m 1 less than or equal to J
• 72:30 - 73:00 less than or equal to n actually okay so this is the transpose okay so let's see if you look at this then we get like let's say if I take a is equal I'll just explain you quickly maybe by taking a 2x3 Matrix a11 a12 a13 okay a21 a22 a33 sorry a23 okay so a 2 3 so let's say if you have a transpose which is equal to B okay I forgot to
• 73:00 - 73:30 write it is denoted by a transpose it is denoted by a with a t on the superscript or sometimes a dash on the top of a a transpose equal to B implies B is equal to a11 a12 a13 a 2 1 a a 22 a 23 correct so
• 73:30 - 74:00 e 23 correct this is what you have yes or no of course if you transpose this Matrix this is what you will get yes will you all agree with me please type in the chat box if you transpose this Matrix this is what you're going to get right this Ser is also for DS and a yes okay we'll talk about partition matrices their properties and all maybe in the third of fourth lectures okay Vector spaces everything so please type in the chat box will you all agree with me
• 74:00 - 74:30 this everyone please type in the chat box if this is a if I write a transpose of this B uh transpose of this Matrix a then this is how I'm going to get the transpose correct now if this matx has to be the transpose of this matx ultimately this A1 2 and this a12 has to be same correct and then this a23 this
• 74:30 - 75:00 a23 has to be same because these are the same elements ultimately I have just made them transpose now this elements I can write it as if this is the size 2x 3 ultimately this is of size 3x2 this Matrix of size 3x2 I can write it as b11 B12 sorry B21 b31 of course then B12 B B 22 B 32 that's what I can write okay because if I want to write this matx B in terms of B elements then
• 75:00 - 75:30 ultimately it got 3x2 so ultimately it should have the row and two columns that's what we know now if you want to equate these two things okay because this is the general form of writing B but this is the transpose which you got from this Matrix if these two has to be equated you will understand something a12 element should be same as B21 a21 should be same as B12 that means in general if you check any of these elements this matx will become the transpose of this Matrix if B21 is same
• 75:30 - 76:00 as a12 a21 is same as B12 and so on so that's how this a i j should be equal to BJ I if this is a12 this should contain B21 exactly at the same spot for the equality of the matrices that's what this statement actually tells you clear yes or no please type in the chat box everyone so please type in the chat box
• 76:00 - 76:30 clear did you understand this initial notation of representation of transpose got it okay now let's understand some things again properties of a transpose which is a bit important properties of
• 76:30 - 77:00 transpose properties of transpose okay so let's see first if you have two matrices A + B whole transpose can be written as a transpose plus b transpose okay so you can just split it because ultimately if you you all know Matrix Edition I guess is there someone who don't know Matrix edition if if you don't know I'll explain isues is it clear do you all know how to do Matrix edition if two Matrix are
• 77:00 - 77:30 given to you of same size of course if they are not of same size you cannot add them but if two matri are of same size do you all know how to add them should we learn that yeah maybe because many times questions in Gate examination if you open pqs you'll see they'll give you uh in the indial form okay IG annotations okay just a minute yeah we have some water so uh do you all know how to add
• 77:30 - 78:00 two matrices if two matrices are given then just adding the two matrices do you know how to do that please type in the chat box okay adding corresponding elements of course okay now let's
• 78:00 - 78:30 see so if you add two matrices and if you take transpose that same as taking transpose and adding them up then if you have K * a whole transpose where K is a scalar K is a scalar for scalar you cannot do a transpose of course and this becomes K * a transpose okay and the third axium you all no if you do transpose even number of times you'll get the same Matrix if you
• 78:30 - 79:00 do odd number of times you'll have the transpose of that matx okay so this is what we have now let's look at some types of matrices first types of matrices okay so okay we have we had some slide here fine we'll put it somewhat forward okay so see here types of matrices so put a heading in your notebook we'll see types
• 79:00 - 79:30 of matrices okay so the first kind that we look at is square Matrix of course very simple definitions Square matx so ultimately what is a square Matrix can anyone tell me what is a square matx of course what is the square Matrix all of you quick a bit
• 79:30 - 80:00 fast what is a square Matrix probabilties of T of Matrix we'll see okay because I just want to teach you multiplication of matrices after that I'll teach you properties of test okay because then it's more useful where diagonal exist on number of equal to number of columns same and columns okay of course so you know a matics a a i j m by
• 80:00 - 80:30 n is set to be a square Matrix a square Matrix if you know m equal to n of course number of for equal to number of colum so ultimately it has a diagonal that exist okay so example if you want I can just give you maybe something like a is equal to a b c
• 80:30 - 81:00 d whatever okay you know basically number of number of columns should be equal yeah m equal to n of course okay then clear I equal to it's not I equal to J M should be equal to n of course okay because I equal to J just gives you one element on the principal diagonal that's it okay then let's see the second form is upper triangular Matrix
• 81:00 - 81:30 can anyone tell me what is an upper triangular matx any idea to anyone of you shashwat Baya anel Sam Thomas sh kotla Manoj Kumar loesh Vista Abdul yashi kusaga every one of you lower side zero yeah IG equal to Z for all I fantastic yeah so let's see a matx I'll just write the definition
• 81:30 - 82:00 first a is set to be an upper triangular Matrix an upper triangular matx if all the lower diagonal elements are zeros if all the lower diagonal
• 82:00 - 82:30 elements are Zer okay or you can also write this I've told you what is a lower diagonal element previously a matrix a equal to a i j is for all 1 less than or equal to I comma J less than or equal to n because if it's not Square matx you cannot Define what is the diagonal okay so that's why so L decomposition method please we have just started matrices
• 82:30 - 83:00 okay we'll go through lot of things we'll see system of equations I'll teach you gos elimination L composition gos Jordan everything okay even gos Seidel okay no issues so look here so the Matrix capital A is equal to a i j for all 1 less than or equal to 1 less than or equal to I Comm less than or equal to n is said to be is set to be an upper triangular matx matx so previously I have told you
• 83:00 - 83:30 when does an element fall under the lower diagonal or below the diagonal you can see if uh I have shown it right just a minute somewhere I have shown so here you can see this denotes column number this denotes row number of course and uh you can see if I is less than J this is called upper diagonal element and if I is greater than J then the element is lower diag element and to call this Matrix an upper triangular Matrix I'm telling all the lower diagonal elements are zero
• 83:30 - 84:00 if a i j is equal to Z for all I greater than J okay so example you can see Y is equal to 2 3 0 4 2 or 0 7 8 0 4 or 0 0 0 you can have some something like this look here if You observe something this
• 84:00 - 84:30 is the diagonal and this is the lower triangular elements and all lower triangular elements okay so you can see all these elements are zeros when I'm defining the statement I never bothered about these elements or this elements only thing that should happen is all my lower element should be zeros it is zeros it's not mandatory that these elements are non zeros or those numbers are not negative you can do it anything okay so
• 84:30 - 85:00 lower is zero yeah basically lower is zero Okay then if you look at upper lower triangular matx then lower triangular matx okay so let's see Now by this time you can understand a matx a is equal to a i j n by
• 85:00 - 85:30 n for all 1 less than or equal to I comma less than or I comma J less than or equal to n is said to be a lower triangular Matrix if a i j is equal to zero for
• 85:30 - 86:00 all I less than J okay that's it means all the upper elements are zeros okay so this tells you this statement which is here this statement tells you all upper diagonal elements are zero all upper diagonal elements are zero of course got it clear and if you want to
• 86:00 - 86:30 look at an example if you take a11 A1 a22 a 2 three then all these elements in the upper triangle should be zero and you have this is what we have here okay soorry this is the y 3 3 this is what we have okay and clearly in this case all the upper
• 86:30 - 87:00 diagonal elements are zeros so this is the upper triangle okay and you guys want to make all A J is equal to Z for I greater than sorry less than J of course okay so this is what we have I have a request in these days these are various new types of questions are coming in Gate from numerical methods would you take care of this why not we'll definitely take okay
• 87:00 - 87:30 so this is a lower triangular Matrix now there comes one special case which is diagonal matrix okay and the Really the use of diagonal matrix is like unlimited okay Frankly Speaking many times even though the Matrix is uh containing all the elements we intentionally make it diagonalized okay that's what we call the diagonalization process we will see diagonalization of matrices maybe in the last lecture of this module linear algebra so I'm telling you making a matrix into a diagonal form could help you a lot actually in solving Solutions
• 87:30 - 88:00 okay we will see when we are looking at the concepts of diagonalizations okay so upper and lower triangular matrices are only for square Matrix or not yes because for other matrices you cannot Define what is a diagonal that's why okay so let's see diagonal matrix diagonal matx okay then let's see a
• 88:00 - 88:30 matx a equal to a i j for 1 less than or equal to I comma J less than or equal to n is set to be a diagonal matrix matx if a i j is equal to Z for all I not equal
• 88:30 - 89:00 to J that's it okay which means so can you please tell where this Matrix is used ear engineering applications like in F and cfd how Matrix is used okay please take this normally in F you calculate stiffness matrices okay so normally you have this if you many of you might have studied your + one you you might have F equal to KX okay so that's in One Direction so let's say if material can actually is elastic in different directions we apply the traction is equal to some K time the displacement okay so there K is actually
• 89:00 - 89:30 a stiffness mattics similarly when you're coming for cfds and all we look at Spar systems okay which are a bit Advanced which which is not in the gate silver scope of course to any branch in fact so to solve this SP systems and all we go for diagonalizations okay and not only SP systems normally if you go for transformations of coordinates okay so let's say if you have uh we'll see much more detailed in Vector calculus if you resolve a vector in XY coordinate system and in polar system how the components of velocity are connected okay how the
• 89:30 - 90:00 components of velocity in cartisian is connected to the components of velocity in polar this connection is again given by matrices so there are plenty of advantages when you deal with matrices we'll see in each uh in each way okay so now you can see what is a diagonal matrix of course only should have diagonal elements okay so Lambda 1 0 0 maybe Z Lambda 2 0 0 0 Lambda 3 let's say these are three elements okay then this is what you have you call it diagonal
• 90:00 - 90:30 matrix if this are the diagonal matrices then definitely upper diagonal element should be zero and lower diagonal element should also be zero correct so if upper diagonal is Z then ultimately a i j is equal to Z if I greater than Z this actually implies a i j equal to Z for all I not equal to J correct because
• 90:30 - 91:00 it's telling you if I is get than Z or if I is less than Z then still this elements are zero that means in a single equation you can write whenever I is not equal to J this elements are of course zero good can diagonal elements also be zero why not see I'm never putting instruction on the diagonal elements I'm telling you these elements are zeros if I not equal to I'm not talking anything about the elements when I equal to J got it they can be Z they can be non- zeros it depends okay Abdul
• 91:00 - 91:30 clear see in life having Clarity is very important okay see basically I'm telling you let's say if you're working in office tomorrow okay let's say for example if I'm your boss and if I tell you uh okay Abdul you go and check whether Vista has come today or not then it doesn't mean that you should go and call vist to me your boss told did he come or not you should just see and say yeah he came because but some people what they do whenever boss says whether did he come then immediately they go there and call that boss is calling you
• 91:30 - 92:00 but this is not the way okay so same thing when I'm telling about non diagonal elements please be focused with the non- diagonal elements that's it okay anything can happen to the diagonal elements and depending upon anything can happen to diagonal elements we will do some analysis again okay so that we will see again when we go for uh calculating determinants okay clear uh voice of sa thank you so this is diagonal element diagonal matrix of course five four okay
• 92:00 - 92:30 five five scalar matx scalar matx so a matx a i Comm than or equal to n is said to be a scalar
• 92:30 - 93:00 matx Scala Maxs if if do anyone know so how long this lecture will be till 9:00 p.m. okay 27 minutes more clear because I understand keeping all this stuff on the same day can actually make you iritated so we will deal in a step Wise Way no issues so can we say that diagonal matx is subset of upper triangular yeah definitely we can say that I'm actually PR but want to learn math from you after
• 93:00 - 93:30 learning your Theo recorded lectures thank you yeah thank you har Krishnan Rajan uh what we talking yeah so scalar mattics yeah IG is equal to K for all equal to J loesh you my student right if I'm not wrong because only you are the one who is telling all the answers are you my student loes just by curios I'm asking somewhere you might have learned math from me right because you're the one only who is all the exact answers yes I know otherwise who else tells that good anyway I'm just kidding anyone can tell that look if a i
• 93:30 - 94:00 j equal to K for all I equal to j z i not equal to J of course okay that means it's special case of diagonal matx where all the diagonal elements are equal that's it okay so case a Scala in general example Y is equal to K 0 0 0 K 0 0 0 K of
• 94:00 - 94:30 course here okay so this is what you get now you can see if k equal to 1 then Y is called you know if you put k equal to 1 then what is a let's see if the constant value is one then what is this Matrix if k value is equal to 1 then what is the matx it's a identity Matrix right yeah of course
• 94:30 - 95:00 unit Matrix then so it's a identity Matrix or it's basically called Unit Matrix so this is identity matx and if k equal to Z do anyone know the name for this if k equal to Z then it is p it's unit Matrix basically if k equal to Z it's called a null Matrix
• 95:00 - 95:30 right null or it's also called z matx z matx okay so this basically a scalar matx now let's see let's go to the definitions of some other matrices symmetric Matrix symic
• 95:30 - 96:00 matx a matx a equal to a i j for all 1 less than or equal to I Comm less than or equal to n is said to be a symmetric Matrix if if
• 96:00 - 96:30 if when do you call the matx as symmetric Matrix transpose of a matrix is itself fine okay two I agree a equal to a transpose okay what can you say about AIG J like we are telling everything in terms of a know so what can you say in terms of a you can think think and tell you can
• 96:30 - 97:00 say yes exactly Satya a i j should be equal to a j i because it should equal its own transpose correct so a i j equal to a j i right a i j equal to a ji I not equal to J if I equal to J also then a i i is equal to a I I know correct yes a Ji Mega very good correct Abdul Satya
• 97:00 - 97:30 everyone so a ji should be equal to a ji and if you just quote an example we have a is equal to a11 a12 a13 for example again a 21 let's say for example if I okay a 21 a22 a23 a 31 a32 and a 3 3 for example okay then ultimately if this Matrix has to be a symmetric Matrix then this element
• 97:30 - 98:00 should be equal to a12 correct because when you do transpose this m this element comes this element in the first or second column comes to the element in the second of First Column so that means when this element comes to this place and if a and a transpose has to be same then this element and this element should be same a12 should be same as a21 similarly this a13 should be same as a31 this is a13 and ultimately this a23 should be same as this guy here so that's what in general a i j should be same as a j i of course clear to
• 98:00 - 98:30 all okay clear guys is this clear to all of you yes or no please type in the chat box chat box okay going good understanding the things only last 20 minutes so please type in the chat box are you all able to understand the
• 98:30 - 99:00 things each and every point in detail okay so let's get into SK symec Matrix skew symmetric Matrix The Matrix is symmetric but with a skew of negativeness that's it okay so let's
• 99:00 - 99:30 see in this skew symic Matrix a i equal to minus AI good nice same as symic Matrix minus will be added sir can you please explain this for all then that inequality being written please sorry I didn't get you so can you please explain this for all then that inequality being written inequality where did inequality I'm not getting did I get any inequality anywhere this
• 99:30 - 100:00 one this INE equality 1 less than or equal to I Comm less than or equal to n that's it okay it tells you like let's say if it tells you 2 less than or equal to X less than or equal to 4 then you'll say x can be 2 3 4 correct or basically you can say x lies in between 2 and 4 okay but here this I comma J are only integers because and column numbers can be only integers so that means this x can take 2 3 4 that's what I'm writing okay clear nothing
• 100:00 - 100:30 bigger than that okay so let's see yeah good a i is equal to minus Aji good of course now people are getting okay initially when I started the class when I'm telling indial notation many of you felt difficulty but now you guys are suggesting me to write AIG equal to minus AI for symmetric CU symc good so yeah of course to a matx capital A is equal to a i j for all
• 100:30 - 101:00 1 less than or equal to I comma J less than or equal to n is said to betic is set to be just a minute okay is set to be skew symmetric if a i j is equal to minus a j I actually okay so a i j is equal to minus Aji so
• 101:00 - 101:30 ultimately you know this statement of course which many of you are familiar this actually tells you that a is equal to minus a transpose or some people they live to say a plus a transpose is equal to Z both are correct now let's observe something few points about this principal diagonal
• 101:30 - 102:00 elements principal diagonal elements of SQ symmetric matx are can anyone tell me what would be the value of a principal diagonal element of a SK symmetric Matrix also a equal to Z for I equal to
• 102:00 - 102:30 J for Q symmet that's what I I just want to see but just tell me if you look at the principal diagonal element of symmetric matx can anyone tell me what what would be the value of that element if a transpose has to be equal to minus a come on all of you quick faster
• 102:30 - 103:00 zero yeah zero okay it is zero zeros I look many of you are thinking just tell me one one way when you transpose a matx will the principal diagonal elements change their position let's say I have a matrix okay now you're transposing that Matrix if you transpose that Matrix then will you or will the principal diagonal elements will they change their
• 103:00 - 103:30 positions if you transpose a matx will the principal diagonal elements change their positions no correct they are not changing so let's say if one of the principal diagonal is X now when you are transposing that the X remains as X in the same position Okay then if you're telling X is equal to- X okay because this minus sign is there okay and whenever the element lies on the principal diagonal it don't changes it position that means a number should be negative of itself if the M if the if the element is on the principal diagonal element correct if the element is on the
• 103:30 - 104:00 principal diagonal element that means the element on the principal diagonal should be equal to negative of itself which number will be equal to negative of itself tell me which number will be equal to negative of its own which number will be equal to negative of its own which number will be negative of its
• 104:00 - 104:30 own don't know xal to- x x value what is the value of x I should ask in a mathematical way no X is equal to- X then what is X then people are happy put that- X on this side and 2x equal to 0 since 2 is not equal to Z so X would be Z that's it right Z so you know since a i j equal to minus a ji for IAL
• 104:30 - 105:00 to J what you can get a i i is equal to minus a i i this implies 2 * of a i i is equal to Z so a i i is equal to0 that's it okay so this tells you okay so every principle diagonal element on atic Matrix has to be zero undoubtedly and so example if I just give you an
• 105:00 - 105:30 example 0 0 0 you fill with zeros if you keep a12 here it should be minus a12 then if you keep a13 here it should be minus a13 then maybe you can keep a 32 minus a A2 that's it okay also I just want to ask you one question before I ask you this question I would like to solve a question okay I want to solve a question come on solve this
• 105:30 - 106:00 question I'm connecting poll of course for previous answer the answer is 144 none of you have answered fortunately good now for this poll I'm giving P of uh you know say 90 seconds start see for Max m m i j for all this is small I of course okay so I comma J equal to 1 2 3 4 so I one last this is same notation as 1 less than or equal to I comma less than or equal 4 the diagonal elements are all zero and m i Jal to minus m i mji of course the
• 106:00 - 106:30 minimum number of elements required to fully specify the matx is so you can actually type the answer in the chat box that automatically comes in the poll okay come on quick so how many elements do you need to completely specify the matx in that matx you have 16 elements elements okay because it's 4x4 so how many elements you have to specify fully if you want to know the complete
• 106:30 - 107:00 Matrix this is actually one year gate question solve
• 107:00 - 107:30 see if your answer is zero please type A in the chat box if your answer is six please type B if your answer is 12 please type c and 16 d kushak p p six okay maybe okay the correct answer is six of course okay 100% uh butaki Atma only has done it fine okay maybe few responses has not taken good see it's not 12 basically I'll tell
• 107:30 - 108:00 you why m i j is equal to minus mji and you know this I comma J are going from 1 to 4 M can be written as M11 M12 I'm just writing so that you can understand the things M21 M22 M23 this is previous year gate question okay you see this this notation is given that's why I'm teaching you
• 108:00 - 108:30 this so look for four okay so see here now whenever I say this m i J is equal to minus mji immediately you can figure out one thing that this is AQ symmetric Matrix of
• 108:30 - 109:00 course SK symetric matx correct now in general if you want to know anything about 4x4 matx I should know all the 16 elements okay I should know what is M11 M12 similarly till m44 I should know each and everything but since this is Q symmetric I need not tell you what are the diagonal elements yes or no let's understand in a step by step way look
• 109:00 - 109:30 since it is Q symmetric I need not tell you what are the diagonal elements you know by default they are zeros I need not tell you this information correct will you all agree if I say if in this matx if this is SK symetric then I need not give you information about the diagonal elements because you know that it is zero
• 109:30 - 110:00 correct yes or no yeah please type in the chat box you guys can just simply sit and type it so please type at least yeah yes now let's say if I give you information about this element then do I need to give you information about this element
• 110:00 - 110:30 if I give you M12 should I again give you what is M21 value if I give you data about M12 then do I need to give you about M21 is it necessary for you to give M21 again if I give M12 already will you cover from g x point of view yeah I'll cover Legend of pubg good that's why I've added sequence and at the last okay so basically what I'm telling is if I give this element I need not
• 110:30 - 111:00 talk about this because if I give M12 you can by default write M21 because if you put M21 M21 equal to if you put IAL 2 and Jal 1 M21 is nothing but minus M12 so if I give this then I need not give this okay because this is just the negative sign of this element similarly if I give this I need not give this if I give this I need not give give this similarly giving M23 need not give these elements if I give m34 I need not give means out of this
• 111:00 - 111:30 remaining 12 elements if I refer any six okay simple negative of this uh this thing correct so if you see out of this total 16 elements these four I need not mention they are zeros by default now out of remaining 12 if I specify this either upper triangular six or lower triangular six then I can get information about the full Matrix okay that's why it is six there are six independent
• 111:30 - 112:00 elements six independent elements and it's a previous gate question answer is B okay so there are six independent elements in that matx of total 16 elements if I can specify those six elements you can 100% tell all the 16 the information about all the 16 so what about the diagonal 1 Z won't that be counted no because if I tell you the Matrix is Q symmetric I need not tell you the diagonal is zero you know that the diagonal is zero by default okay so diagonal is not counted
• 112:00 - 112:30 because I need not tell you that diagonal is zero you can identify it by yourself if I tell you the Matrix is a extic matrix okay because of this what I have taught you here you know I have taught something here so uh where is it yeah here okay so this two steps I have taught you okay we'll solve one more question which which could give some important things for an N byn array which is n byn Matrix of course V is
• 112:30 - 113:00 defined as follows v i j is equal to IUS J for all I comma J 1 for all I comma such that 1 less than or equal to I Comm less than or equal to n then the sum of elements of the array V is Dash which is nothing but the same question v i j is equal to IUS J for all 1 less than or equal to I comma J less than or
• 113:00 - 113:30 equal to n then sum of all elements of V of V can anyone tell me if the element in see for example if I have defined I plus J okay you have seen how things come up if it is IUS J then what would be sum of all the
• 113:30 - 114:00 elements of the matx okay I'm again connecting pole so single Choice I'm giving maybe 60 90 seconds you take okay fine no issue think see if you want to enter zero you enter a okay if you want to enter n you enter
• 114:00 - 114:30 B this is n plus1 then you enter c this time many of you are failing to answer okay good yay oh so many typing a
• 114:30 - 115:00 maybe maybe this is the last question that we solve today because only 5 minutes is left corresponding elements will cancel out and diagonal will be zero good logic of course see once you participate at the polls as I told you you can check it in
• 115:00 - 115:30 unity app okay or basically in the unity website so there you can check okay so yeah the correct answer is zero of course oh there's a big list nice Abdul Vista it takes only top five I guess I'm not sure but yeah hi kishna good so let's see first of all I want just tell you one thing v i j is equal to IUS J correct this implies if I talk about VJ I this should be J minus I
• 115:30 - 116:00 yes or no correct because if it is I J it is IUS J so if it is j i it should be J minus I of course okay so if you take a minus sign it becomes IUS J and IUS J can be replaced with i j so BJ I is equal to minus v i j this tells you the Matrix is Q symmetric matx
• 116:00 - 116:30 is cutic okay matx istic and you know sum of all the elements of SQ symmetric matx of SQ stic matx is zero here is the correct answer okay
• 116:30 - 117:00 so this can be done even easily look for example if I look here okay these three elements do not contribute for any summation because they are zeros itself some of these two would be zero some of these two would be zero some of these two would also be zero okay so for every element there will be a contradicting element with a minus sign so sum of those two elements would be zero like for example you see I'll just give us a point here you can write sum of all the
• 117:00 - 117:30 elements of stic matx of SQ symmetric matx are sorry sum okay is is zero like example you can see you know one thing that sum of summation I = to 1 to n summation J = 1
• 117:30 - 118:00 to n a i j this is what you have okay so this is summation of course just let me tell you one thing let's say this is equal to K for example okay then can I write this as can I write the same summation as if I interchange I and J of course this can be written as K is equal to summation J = 1 to n Su summation I =
• 118:00 - 118:30 1 to n a j i okay but this Aji can be written as which is equal to minus K of course because these summations can be flapped so k equal to minus k that's it okay clear so let's say initially you have take the sum k then you can
• 118:30 - 119:00 interchange these two of course summations and Aji so Aji can be replaced with minus a i j so if you take this minus and again replace this this is minus K so k equal to Z clear not clear but how many of you it is not clear tell honestly I'll explain step by step no issue for how many of you it is not clear please please say honestly okay be honest I honest to you so you two be honest to me that's it not clear good I'll explain it
• 119:00 - 119:30 okay good okay yeah I'll explain again no issue see here sum of all the elements see first of all you know a i j is equal to minus a j i this implies a i j plus a ji is equal to Zer this is what you know okay
• 119:30 - 120:00 a i j plus Aji is equal to zero uh so any upcoming English batch yeah there are lot of uh coming up every uh I think every month some orientation sessions goes on and you can uh look at gate W mechanical English Channel of course the main channel of mechanical uh EC tily these kind of channels for there so you can look at there you definitely you'll be provided with the updates okay then let's see and once again I'm telling you this is my telegram group so if some of you are interested you can join there if you want to get in touch
• 120:00 - 120:30 with me and if you want to know all the updates about the English Channel then definitely please go for this group so I have clearly kept this is gatea English and this is uh telegram uh channel of mine okay so anyway let's move forward to this after this we'll close this lecture today so let's say you have so let's say sum of all the elements sum of all the elements all the
• 120:30 - 121:00 elements summation I = 1 to n summation J = 1 to n a i j this is what you'll get I have told you already okay and I have told you how this summation could actually give you the complete sum of all the elements I have explain you now let's see this expression if you uh just expand for example you will have sum of all the elements a11 plus a12 + A2 1 sorry
• 121:00 - 121:30 a13 and so on plus A1 n plus a21 plus a22 plus a23 plus and so on a21 plus a31 plus a32 plus a33 plus and so on a3n Plus if this goes on you'll have a N1 a N2 a
• 121:30 - 122:00 N3 okay will you all agree class will be daily okay daily from 7:00 p.m. to 9:00 p.m. of course we'll go to the details just before I close so this is what we have correct so this is going to be the sum of all the elements of that Matrix yes yes or no please type in the chat box this is going to be the sum of all the elements of the matx which is of course the same as this
• 122:00 - 122:30 okay now what I'm telling is this elements are zeros by default a220 because this is Q symmetric now since you know a i plus aj1 equal to Z this tells you a11 + A2 1 sorry a12 + A2 1 this is equal to Zer similarly this element will cancel this this element will cancel this because A2 3 is equal to- a32 when you add these two elements gets cancel similarly A1 n and a N1 A2 n a N2 because a N2 will be
• 122:30 - 123:00 minus of this number when you add A2 n to this you'll get zero this and finally everything gets cancelled and you'll get zero actually okay this is what I have done in a slight short way okay but anyway so in AC math syllabus do we have transform the it's not included in I syllabus it's actually there because you know basically if you want to work at your Control Systems you need lapas transforms okay and if you're working with signal trans systems you'll work with foure transforms so ultimately indirectly may not be mentioned in the
• 123:00 - 123:30 math syllabus expecting maybe signals and systems faculty to teach that but you have to have good knowledge of laas and for transforms okay so how you showed the solution in symbolic manner earlier basically what I have told is I have assumed this total sum to be K okay then I have changed the indices so this becomes a21 and this becomes a12 this becomes a31 and this becomes a13 so this total summation is also equal to K okay
• 123:30 - 124:00 but when I have changed the signs I can replace a12 with minus a21 a31 with minus a13 so like that I can take one minus sign common from all these numbers okay I'll just tell you for example if I take a 2X two okay you'll understand maybe easily so let's say this is a21 for example initially what I have taken is summation K is equal to a12 + A2 1 of course okay
• 124:00 - 124:30 now this since I have changed the indes J and I I have interchanged I can write this as a21 + a12 because these two are same but actually I can replace this a21 as minus a12 minus A2 1 if I take minus sign common I'll have a12 plus A2 1 and this term can be replaced again with this k so minus K so k equal to minus K so K has to be zero this is what I have done in a shortcut that's it did you
• 124:30 - 125:00 understand first thing I have done is I have interchanged the notations and then I have used this minus sign property so to show k equal to minus K so ultimately K has to be zero that's it that's what I have done okay but I think people are familiar with this and happy with this okay no issue okay did you understand
• 125:00 - 125:30 this is this clear what I have done I have taken the sum as K of course diagonal elements are any zero I have neglected them or even if you take a I I will become zero of course so first I have flipped the indices so A2 1 plus a12 this summation and this summation will remain same but here here I have used the property of stic so I got negative of this K actually so k equal to minus K so K is zero that's it okay A bit of
• 125:30 - 126:00 smarter but actually you know uh we when we are doing programming especially this kind of stuff helps you a lot fine so anyway we'll stop here for today because I know for some of you who have learned this for the first time it would be slightly hectic so I'll stop here for today so let us have a quick Rie so basically these are the channels if you wish you can join in these two so you can get some understanding then this is the syllabus that we are going to cover we started with linear algebra
• 126:00 - 126:30 okay then these are the books of course this is for conceptual and problems and this for technical connectivity practical connectivity of the subject and so uh we have started with matx definition then we have went to a lot of things fortunately and we have talked about types of matrices upper triangular lower triangular then we talked about some graphs to make you understand few nice things then properties of transpose and scalar matx principal
• 126:30 - 127:00 diagonals okay then this summation then we have solved some two problems of course we'll solve more in the upcoming classes so tomorrow's class we will be mainly dedicated to Matrix algebra okay we'll see properties of addition subtractions multiplications again in multiplications we have some nice uh things to understand so we will see all these things in tomorrow session and then uh we will progress slowly okay so again I would like to remember you please go go for this link okay unity. pw. live so that you can track all your record at one place okay even if you
• 127:00 - 127:30 forget them at later stage you can still have the resource in this and whenever you're typing doubts I suggest you please type this exclamation doubt and then type doubts and similarly if you want to make a note please type exclamation note and then type whatever you want to save in your uh catalog and of course every day we'll be starting at 7:00 p.m. I hope I didn't bore you much today so 7:00 p.m. Monday to Saturday and this fellow is me and yeah in English only I'll be teaching in this complete uh batch okay so in English we will teach and classes will be from
• 127:30 - 128:00 Monday to Saturday 7:00 p.m. and as I told you this will be a complete comprehensive course okay I'll deal each and every point and if you have doubts you can always ask me you open fre okay and at some point in the day time if you have doubt you can post it in the telegram group people will be happy to answer that okay and yeah I think then that's it from my side you have anything to say fine
• 128:00 - 128:30 then did you all understand each and every Point shash Bay har Krishan digital Battle Grounds okay nice name like the lecture goodi great okay so it's not the first lecture since uh I mean I've been teaching this story for past seven seven plus years so no issues I'll be teaching you in a good way you can trust me and I hope that's what happened in the first lecture okay so tomorrow sh 7 p.m. you come we'll play with some matrices tomorrow also
• 128:30 - 129:00 and then slowly we'll go into the depths okay of determinant system of equations Rank and then after we'll speak about uh system of equations I values I vectors physically geometrically we'll see what is I value what is I Vector we we'll talk about all these things okay so fine if you have any queries you can ask me or else signing off okay I hope all of you like the session liking uh is again I always not hitting the like button but the content what I have delivered okay so thank you all
• 129:00 - 129:30 we'll meet tomorrow at 7: p.m. for matx algebra thank you okay signing off bye good night take care