Adjoint of a Matrix

Duration: 8 min

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  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
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  3. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Probability, Statistics, Signal Space and Optimum Detection, Probability, Statistics
  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. Programming & Data Structures: 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, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  10. Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
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  13. Full Length Mock Tests:
  14. Sectional Tests:
  15. Paper 2 - Full-length Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This linear algebra lecture focuses on the Adjoint and Inverse of a matrix, specifically deriving the formula for the inverse using the adjoint. The instructor defines the adjoint as the transpose of the cofactor matrix ($Adj(A) = C^T$) and establishes the fundamental property $A \cdot adj(A) = |A| \cdot I$. Through algebraic manipulation, he derives the inverse formula $A^{-1} = rac{adj(A)}{|A|}$. The lecture concludes by explaining that if the determinant $|A|$ is zero, the matrix is singular and the inverse is undefined, marking a critical condition for matrix invertibility.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card and a visual of a 2x2 determinant formula ($ab - cd$). The instructor introduces the topic "ADJOINT/ADJUGATE/ADJUNCT" and writes the definition $Adj(A) = C^T$, clarifying that the adjoint is the transpose of the cofactor matrix. He then writes the first major property on the board: $A \cdot adj(A) = |A| \cdot I$. The text "Adjoint is the transpose of cofactor matrix" appears in brackets next to the formula.

  2. 2:00 – 5:00 02:00-05:00

    The instructor derives the inverse formula by manipulating the property $A \cdot adj(A) = |A| \cdot I$. He divides the equation by the scalar determinant $|A|$ to get $A \cdot rac{adj(A)}{|A|} = I$. By comparing this to the definition $A \cdot A^{-1} = I$, he boxes the final formula $A^{-1} = rac{adj(A)}{|A|}$. The board also displays "also $A \cdot A^{-1} = I$" to show the comparison. He emphasizes that if $|A| = 0$, the inverse is not defined, writing "If $|A| = 0$, then $A^{-1}$ is not defined" on the board.

  3. 5:00 – 8:06 05:00-08:06

    The instructor re-derives the steps on the right side of the board to reinforce the logic, writing $A \cdot rac{1}{|A|} adj(A) = I$. He discusses scalar multiplication properties like $x(M_1 M_2)$ to justify the algebraic steps involving the scalar $1/|A|$. The text "Singular" is written next to the condition $|A| = 0$ and the inverse symbol is crossed out to indicate non-existence. He also writes "If $|A| = 0$, then $A$ is not invertible" to reinforce the concept. Finally, he summarizes the dimensional relationship: $A_{n imes n} o adj(A) o C^T$, showing how the matrix transforms into its adjoint.

The lecture provides a complete derivation of the matrix inverse using the adjoint method. It connects the definition of the adjoint to the fundamental identity involving the determinant, leading to the standard inverse formula. The key distinction made is between invertible matrices (where $|A| eq 0$) and singular matrices (where $|A| = 0$), ensuring students understand the limitations of the inverse operation and the conditions required for a matrix to be invertible. The instructor uses clear board work to show the step-by-step algebraic division.

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