Echelon Form - Gauss Elimination Method (Solving Equations)

Duration: 9 min

This video lesson is available to enrolled students.

Return to /learn/ZERO-TO-HERO/aptitude/linear-algebra/system-of-linear-equations/asset-echelon-form-gauss-elimination-method-solving-equations after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. Database Management System: 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
  2. Discrete Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic
  3. Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  4. Computer Architecture: Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design
  5. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Deadlock, Virtual Memory, Disc Scheduling, File Management, Memory Management, Threads & Process Creation
  6. Computer Networks: Introduction to CN, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Network Security, Hardware Basics
  7. Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  8. Algorithms: Algorithm Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Time Complexity Analysis
  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
  11. C language: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums
  12. Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  13. Emerging Trends & AI: Emerging Tech Trends, Artificial Intelligence
  14. Computer Graphics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing
  15. Aptitude: Percentage, Ratio and Proportion (Ratios), Logarithm, HCF LCM, Divisibility Rules, Average, Sequence and Series (Series), Number System, Powers and Exponents (Surds and Indices), Algebra, Permutation and Combination, Probability, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Pipe and Cistern, Speed Time and Distance, Mixture and Alligation, Mensuration and Geometry, Mental Ability, Age Problems, Partnership, Statistics, Data Interpretation, Arithmetic, Speed Mathematics and Basic Mathematics, TCS NQT Advanced Quantitative Ability, Gamified / Game Based Aptitude, Linear Algebra, Trigonometry
  16. Reasoning: Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Data Sufficiency, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Seating Arrangements, Blood Relations, Directions (Direction Test), Analytical Reasoning (Counting Figures Reasoning), Puzzle Solving (Puzzles), Calendar, Clock, Cubes & Dices, Selection Decision Table (Decision Making), Logical Sequence Of Words, Crypt Arithmetic, Venn Diagram, Data Arrangement, Inequality, Analogy, Ranking, Order and Sequence, TCS NQT Advanced Reasoning Ability
  17. English (Verbal Ability): Vocabulary, Noun, Pronoun, Adjectives, Preposition, Articles, Direct and Indirect Speech, Tenses, Conjunction, Subject Verb Agreement (Verb Noun Agreement), Modals, Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Cloze Test (Cloze Passage), Verbal Analogy (Word Based Analogy), Sentence Construction, Active Passive Voice (Voice), Determiner, Facts, Inference and Judgements, Adverb, Interjection, Verb, English Writing Ability
  18. Engineering Mathematics: Calculus, Vectors
  19. GATE Live Class Recordings 2026: GATE 2026 Live Class
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video presents a detailed lecture on Linear Algebra, specifically focusing on the Gauss Elimination Method for solving systems of linear equations. The instructor, Yash Jain from Knowledge Gate, guides viewers through the process of converting a system of three linear equations into an augmented matrix. He demonstrates the systematic application of row operations to transform the matrix into an upper triangular echelon form. The lesson concludes with a clear explanation of back-substitution to find the values of the variables x, y, and z, providing a complete worked example for students to follow.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video begins with a brief title card displaying determinant formulas for 2x2 and 3x3 matrices before transitioning to the main lecture. The instructor, Yash Jain, stands before a whiteboard titled "Gauss Elimination Method". He writes a system of three linear equations: x + 3y - 2z = 5 (labeled 1), 3x + 5y + 6z = 7 (labeled 2), and 2x + 4y + 3z = 8 (labeled 3). He explains that the coefficients of the equations and the constants are written in a single matrix known as an "Augmented Matrix". He writes out the 3x4 augmented matrix: [1 3 -2 | 5; 3 5 6 | 7; 2 4 3 | 8]. He notes that the goal is to "try converting it to echelon form". Below the matrix, he labels the left part as the "coefficient matrix" and the right part as the "constant matrix". He also writes the general form Ax = b and the augmented matrix notation [A|b] at the top right.

  2. 2:00 – 5:00 02:00-05:00

    The instructor begins the row reduction process to eliminate variables. He labels the rows R1, R2, R3. To eliminate the x-term in the second row, he performs the operation R2 <- R2 - 3R1. He writes out the arithmetic on the board: 3-3, 5-9, 6+6, and 7-15. The new second row becomes [0 -4 12 -8]. Next, he eliminates the x-term in the third row using the operation R3 <- R3 - 2R1. He calculates the values: 2-2, 4-6, 3+4, and 8-10. The third row becomes [0 -2 7 -2]. The matrix is now partially reduced, with zeros in the first column below the pivot. He circles the numbers being operated on in red to highlight the calculation steps. He writes the operation R2 + R2 - 3R1 on the left side of the board to indicate the transformation.

  3. 5:00 – 9:11 05:00-09:11

    The instructor continues reducing the matrix to achieve a diagonal form. He focuses on the second column, aiming to eliminate the -2 in the third row using the -4 in the second row. He performs the operation R3 <- R3 - 1/2 R2. The calculations yield: -2 - (-2) = 0, 7 - 6 = 1, and -2 - (-4) = 2. The final row is [0 0 1 2]. The matrix is now in echelon form. He then solves for the variables using back-substitution. From the last row, 1z = 2, so z = 2. Substituting z into the second row equation -4y + 12z = -8, he writes -4y + 24 = -8, which simplifies to -4y = -32, so y = 8. Finally, substituting y and z into the first row equation x + 3y - 2z = 5, he calculates x + 24 - 4 = 5, resulting in x = -15. The final solution is presented as x = -15, y = 8, z = 2, with the answers circled in red. He writes the intermediate equations like 0x + 0y + 1z = 2 to show the logic clearly. The video ends with a "Thanks for watching" screen.

The lecture provides a comprehensive, step-by-step demonstration of solving a system of linear equations using the Gauss Elimination Method. It covers the initial setup of the augmented matrix, the application of elementary row operations to reach echelon form, and the final back-substitution to determine the unique solution.

Loading lesson…