Properties of Adjoint, Inverse & Determinants

Duration: 8 min

This video lesson is available to enrolled students.

Return to /learn/DSSSB-TGT-COMPUTER-SCIENCE-PAPER-2/mathematics-i-ii-iii-iv/linear-algebra/adjoint-and-inverse-of-a-matrix/asset-properties-of-adjoint-inverse-determinants after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. Programming, Data Structures, OOP & Java: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Java, OOPS with C++, Java Fundamentals, Array, String, IO & References, Language Design, Intro to Compilers, Visual Basics, Syntax Analysis: Bottom-Up, Object Oriented Technology
  3. Computer Networks, TCP/IP & Network Security: Introduction to CN, Data Communication, DLL: Flow Control, DLL: Access Control, DLL: Error Control, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics, Network Access Technologies, Network Security, Society, Law & Ethics, Cloud Computing, Mobile Computing, Net Layer:Routing Protocol, Cyber Safety and Security
  4. Teaching Methodology / शिक्षण विधि: Growth and Development / वृद्धि और विकास, Concept of Pedagogy / शिक्षाशास्त्र की संकल्पना, Theories of Development / विकास के सिद्धांत, Inclusive Education / समावेशी शिक्षा, Experiential Pedagogy / अनुभवात्मक शिक्षाशास्त्र, Assessment and Evaluation / आकलन और मूल्यांकन, Socialisation and Gender / समाजीकरण और जेंडर, National Education Policy (NEP) 2020 / राष्ट्रीय शिक्षा नीति (NEP) 2020, Cognitive Processes / संज्ञानात्मक प्रक्रियाएँ, Teaching Planning and Teaching Skills / शिक्षण योजना और शिक्षण कौशल, Personality Development / व्यक्तित्व विकास, Teaching Methods / शिक्षण विधियाँ, Learning and Behavioural Theories / अधिगम और व्यवहारवादी सिद्धांत, Learners & Learning Process / अधिगमकर्ता और अधिगम प्रक्रिया, Factors Affecting Teaching / शिक्षण को प्रभावित करने वाले कारक, Teaching Methods and Approaches / शिक्षण विधियाँ एवं उपागम, Teaching-Learning Process and Classroom Management / शिक्षण-अधिगम प्रक्रिया एवं कक्षा प्रबंधन, Teaching Aids and Educational Technology / शिक्षण सहायक सामग्री एवं शैक्षिक तकनीकी, Microteaching and Classroom Communication / सूक्ष्म शिक्षण एवं कक्षा संप्रेषण, Assessment and Evaluation / आकलन एवं मूल्यांकन, Curriculum, Arts and Co-curricular Areas / पाठ्यचर्या, कला एवं सह-शैक्षिक क्षेत्र, Education of Children with Disabilities / दिव्यांग बच्चों की शिक्षा, Teacher: Qualities, Professional Ethics and Development / शिक्षक: गुण, व्यावसायिक नैतिकता एवं विकास
  5. Digital Electronics / Boolean Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Analog-Digital Conversion
  6. Operating Systems & Linux: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, File Management, Disc Scheduling, Unix OS, Linux OS, Distributed Systems, Windows OS
  7. Data Base Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, Relational Algebra, SQL, File Organization & Indexing, Transaction Management, Concurrency Control, Database Security & Authorization, Database Recovery, Data Warehousing & Mining
  8. Internet & Web Technologies: Website, Browser & Services, Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, HTML Basics, CSS, Emerging Technologies, JavaScript-Basics, .Net Programming, Electronic Data Interchange and E-Commerce, XML
  9. Computer Fundamentals, IT Basics & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer, Terms & Abbreviations, MS Office Applications, Instr Formats & Modes, Floating Point Rep, Computer Fundamentals, Cyber Safety and Security, Computer Software, Operating System, Word Processing (MS Word), Computer Networks, Presentation Software (MS PowerPoint)
  10. Python Language: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  11. Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, extras, Fundamentals of Systems and Information Systems
  12. Design & Analysis of Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  13. Mathematics - I, II, III, IV / गणित - I, II, III, IV: Permutation and Combination / क्रमचय और संचय, Logarithm / लघुगणक, Linear Algebra / रैखिक बीजगणित, Algebra / बीजगणित, Calculus / कलन, Trigonometry / त्रिकोणमिति, Vectors / सदिश, Set Theory, Functions, Relations, Propositional and Predicate Logic, Research Process Steps / शोध प्रक्रिया के चरण, Graph Theory, Sets, Relations, Functions and Discrete Mathematics / समुच्चय, संबंध, फलन और विविक्त गणित, Number Systems and Number Theory / संख्या पद्धति और संख्या सिद्धांत, Euclidean Geometry and Mensuration / यूक्लिडीय ज्यामिति और क्षेत्रमिति, Statistics and Probability / सांख्यिकी और प्रायिकता, Coordinate Geometry, Vectors and 3D Geometry / निर्देशांक ज्यामिति, सदिश और त्रिविमीय ज्यामिति, Polynomials, Equations and Complex Numbers / बहुपद, समीकरण और सम्मिश्र संख्याएँ, Plane Mensuration / समतल क्षेत्रमिति, Sampling / प्रतिचयन, Percentage / प्रतिशत, Simplification, Indices and Surds / सरलीकरण, घातांक और करणी, Ratio and Proportion / अनुपात और समानुपात
  14. Computer Graphics, Multimedia & Mobile Computing: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing, Multimedia
  15. Emerging Trends: Emerging Tech Trends, Knowledge Representation, Natural Language Processing, Trade, Marketing and Consumer Protection / व्यापार, विपणन एवं उपभोक्ता संरक्षण
  16. Statistical Techniques / सांख्यिकीय तकनीकें: Probability / प्रायिकता, Statistics / सांख्यिकी
  17. Interpolation - DI / अंतर्वेशन - DI: Data Interpretation / आँकड़ा निर्वचन
  18. Basis of Physics / भौतिकी का आधार: Scientific Achievements and Research / वैज्ञानिक उपलब्धियाँ और अनुसंधान
  19. Previous Year Paper:
  20. Business Studies and Management / व्यवसाय अध्ययन एवं प्रबंधन: Business Communication / व्यावसायिक संचार, Principles and Functions of Management / प्रबंध के सिद्धांत एवं कार्य, Accounting Fundamentals and Recording of Transactions / लेखांकन के आधार एवं लेन-देनों का अभिलेखन, Business Economics / व्यावसायिक अर्थशास्त्र, Depreciation, Final Accounts and Computerised Accounting / ह्रास, अंतिम खाते व कंप्यूटरीकृत लेखांकन, Knowledge Management & New Economy / ज्ञान प्रबंधन और नई अर्थव्यवस्था
  21. MIS / DSS / Expert Systems: Information Systems
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This video lecture focuses on the properties of adjoint and inverse matrices within the context of linear algebra. The instructor begins by listing fundamental properties on a whiteboard, including the relationship between the determinant of a matrix and its inverse. He then provides a step-by-step derivation for the determinant of the adjoint matrix, utilizing scalar multiplication rules. The lesson concludes by cataloging additional properties involving matrix products, powers, scalar multiplication, and transposes, supported by a 2x2 matrix example.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card 'LINEAR ALGEBRA' and a 2x2 determinant formula. The scene shifts to a whiteboard titled 'PROPERTIES OF ADJOINT & INVERSE'. The instructor introduces the first property: $|A^{-1}| = rac{1}{|A|} = |A|^{-1}$, illustrating it with the example $1/2 = 2^{-1}$. He then states that for a singular matrix where $|A|=0$, the inverse $A^{-1}$ does not exist, marking this with a check. He introduces the third property: $|adj(A)| = |A|^{n-1}$ for an $n imes n$ matrix. He extends this to iterated adjoints, writing $|adj(adj(A))| = |A|^{(n-1)^2}$ and $|adj(adj(adj(A)))| = |A|^{(n-1)^3}$. He points to the first property formula while explaining. He also circles the term $|A|$ in the denominator.

  2. 2:00 – 5:00 02:00-05:00

    The instructor moves to a derivation on the left side of the board to prove $|adj(A)| = |A|^{n-1}$. He starts with the definition $A^{-1} = rac{adj(A)}{|A|}$ and rearranges it to $|A|A^{-1} = adj(A)$. Taking the determinant of both sides gives $||A|A^{-1}| = |adj(A)|$. He applies the scalar multiplication property $|kA| = k^n|A|$, where $k=|A|$, resulting in $|A|^n |A^{-1}| = |adj(A)|$. Substituting $|A^{-1}| = 1/|A|$, he simplifies the left side to $|A|^n \cdot rac{1}{|A|} = |A|^{n-1}$, confirming the property. On the right side, he derives the formula for the determinant of the adjoint of the adjoint. Letting $adj(A) = B$, he uses the property $|adj(B)| = |B|^{n-1}$. Substituting back, he gets $|adj(adj(A))| = |adj(A)|^{n-1}$. Finally, substituting the previous result, he arrives at $|adj(adj(A))| = (|A|^{n-1})^{n-1} = |A|^{(n-1)^2}$. He notes that similarly, the triple adjoint yields $|A|^{(n-1)^3}$. He writes 'Similarly' to indicate the pattern continues.

  3. 5:00 – 8:15 05:00-08:15

    The final section lists additional properties of determinants and adjoints. Property 4 states $|AB| = |A||B|$, and Property 5 states $|A^k| = |A|^k$. He lists Property 6 as $adj(0) = 0$ (zero matrix) and Property 7 as $adj(I) = I$. Property 8 is $adj(kA) = k^{n-1} adj(A)$, and Property 9 is $adj(A^T) = (adj(A))^T$. On the right side, he displays a 2x2 matrix example $A = egin{bmatrix} a & b \ c & d \end{bmatrix}$, showing its adjoint $adj(A) = egin{bmatrix} d & -b \ -c & a \end{bmatrix}$ and inverse $A^{-1} = rac{1}{ad-bc} egin{bmatrix} d & -b \ -c & a \end{bmatrix}$. He points to the formula $|AB| = |A||B|$ and the scalar multiplication property $adj(kA) = k^{n-1} adj(A)$ while explaining. The video concludes with these fundamental properties written on the board. He also circles 'mxn matrix' in the previous section.

The lecture systematically builds understanding of matrix properties, starting with the relationship between a matrix and its inverse determinant. It then rigorously derives the determinant of the adjoint matrix using scalar multiplication rules. The lesson concludes by cataloging essential properties for products, powers, and transposes, providing a comprehensive reference for solving linear algebra problems involving determinants and adjoints.

Loading lesson…