Gauss Jordan Elimination & Reduced Row Echelon Form

The Organic Chemistry Tutor
18 Feb 201810:51
EducationalLearning
32 Likes 10 Comments

TLDRThis video tutorial demonstrates the process of solving a system of linear equations with three variables using the Gauss-Jordan elimination method. The instructor meticulously converts the system into an augmented matrix and performs a series of elementary row operations to transform it into reduced row echelon form. Through each step, the video clarifies how to manipulate the matrix to isolate the variables and ultimately determine their values, resulting in the solution: x = 6, y = -1, and z = -2.

Takeaways
  • 📚 The lesson focuses on using Gauss-Jordan elimination to solve a system of linear equations with three variables.
  • 🔢 The initial step is to convert the system of equations into an augmented matrix, placing the coefficients on the left and constants on the right.
  • ➡️ The goal of the process is to transform the matrix into reduced row echelon form, where the main diagonal consists of ones and the rows above it are zeros.
  • 📝 The first row operation involves subtracting row1 from row2 and applying the same to row3 to start eliminating variables from the first column.
  • 🔄 Further row operations include adding and subtracting multiples of rows to manipulate the coefficients and constants, aiming for zeros in the correct positions.
  • 🎯 After performing the necessary row operations, the final row echelon form reveals the values of x, y, and z, which are the solutions to the system of equations.
  • 👉 The script provides a step-by-step walkthrough of the row operations, including the calculations and the reasoning behind each step.
  • 🧠 It emphasizes the importance of careful calculation, as even a small mistake can lead to incorrect results.
  • 📌 The final solution is presented as a set of individual values for x, y, and z, derived from the transformed matrix.
  • 📈 The Gauss-Jordan elimination method is a powerful tool for solving systems of linear equations and is demonstrated to be effective through this example.
  • 🔑 The lesson serves as a comprehensive guide for anyone looking to understand and apply the Gauss-Jordan elimination technique to solve similar problems.
Q & A

  • What method is used in the lesson to solve the system of equations?

    -Gauss-Jordan elimination is used to solve the system of equations with three variables.

  • How many variables and equations are in the given system?

    -There are three variables (x, y, z) and three equations in the system.

  • What is the augmented matrix for the given system of equations?

    -The augmented matrix is: [1 1 -1 | 7; 1 -1 2 | 3; 2 1 1 | 9].

  • What is the goal of the Gauss-Jordan elimination method?

    -The goal is to transform the matrix into reduced row echelon form, where the main diagonal consists of ones and the rows above the main diagonal consist of zeros.

  • How does the process start to eliminate variables from a specific column?

    -The process starts by performing elementary row operations, such as subtracting one row from another, to create zeros in the columns below the main diagonal.

  • What row operation is performed to make the number in the second position of the second row a zero?

    -Negative 2 times the first row is added to the second row to make the number in the second position a zero.

  • How is the mistake discovered in the video script corrected?

    -The mistake is discovered when the instructor realizes that the number negative 6 was incorrectly copied as 4. The correct value is then used to proceed with the calculations.

  • What is the final solution for the system of equations?

    -The final solution is x = 6, y = -1, and z = -2.

  • How are the final values of x, y, and z determined?

    -The final values are determined by looking at the main diagonal of the reduced row echelon form matrix, where each value represents the solution for the corresponding variable.

  • What is the significance of the reduced row echelon form in solving systems of linear equations?

    -The reduced row echelon form is significant because it simplifies the system of equations to a point where the solution can be easily read off from the matrix, indicating the values of the variables.

  • What does the process of transforming the matrix teach about solving systems of linear equations?

    -The process teaches that through a series of systematic row operations, one can manipulate the coefficients of the variables to isolate them and find their values, which is essential for solving systems of linear equations.

Outlines
00:00
📚 Introduction to Gauss-Jordan Elimination

The paragraph begins with an introduction to the Gauss-Jordan elimination method for solving a system of linear equations with three variables. The problem is presented, which involves three equations: x + y - z = 7, x - y + 2z = 3, and 2x + y + z = 9. The process starts by converting the system into an augmented matrix, with the coefficients of the variables and the constants on the right side. The goal is to transform this matrix into reduced row echelon form, where the main diagonal consists of ones and the other elements above and below are zeros. The first step involves elementary row operations to eliminate the first variable from the second and third equations, setting up the stage for further simplification.

05:01
🔄 Performing Row Operations

This paragraph details the row operations performed to simplify the augmented matrix. The process involves subtracting one row from another to create zeros in specific positions. The first operation is subtracting row 1 from row 2, which results in changes in the first two columns but leaves the third row unchanged. Another operation involves adding a multiple of row 1 to row 3 to eliminate the second element in the third row. The paragraph then describes the next steps, which include making the second element of the first row zero by adding a multiple of row 2 to row 1 and making the second row's third element zero by adding a multiple of row 3 to row 2. The aim is to get the matrix closer to the reduced row echelon form.

10:01
📈 Achieving Reduced Row Echelon Form and Solving the System

The final paragraph focuses on achieving the reduced row echelon form and solving the system of equations. The process involves making the remaining elements in the first row zero and then using back-substitution to find the values of the variables. The paragraph describes the operations needed to make the second and third elements of the second row zero, leading to a simpler matrix. Once the matrix is in the desired form, the values of x, y, and z are determined by the elements in the last column. The solution is found to be x = 6, y = -1, and z = -2, which is presented as the final answer to the system of equations.

Mindmap
Keywords
💡Gauss-Jordan Elimination
Gauss-Jordan Elimination is a method used to solve systems of linear equations by transforming the augmented matrix associated with the system into a form known as reduced row echelon form. This process involves a series of row operations that simplify the matrix, making it easier to read and interpret the solutions for the variables involved. In the video, this method is applied to a system of three equations with three variables, x, y, and z, to find their values.
💡Augmented Matrix
An augmented matrix is a mathematical structure that combines the coefficient matrix of a system of linear equations with the constant terms, separated by a vertical bar. It serves as a convenient way to visualize and manipulate the system of equations, allowing for the application of row operations to find the solution. In the context of the video, the augmented matrix is constructed from the coefficients and constants of the given equations to apply Gauss-Jordan Elimination.
💡Row Operations
Row operations are a set of basic operations performed on rows of a matrix during the process of matrix manipulation, such as in Gauss-Jordan Elimination. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another. The goal is to transform the matrix into a form where it becomes easier to solve the system of equations. In the video, various row operations are applied to the augmented matrix to achieve the reduced row echelon form.
💡Reduced Row Echelon Form
Reduced row echelon form is a specific arrangement of rows in a matrix where each leading coefficient (the first non-zero number from the left, also known as the pivot) is 1, and all other elements in the same column as the pivot are 0. This form of the matrix simplifies the process of reading the solution to the system of linear equations, as each row represents a single equation with the corresponding variable isolated on one side. The video demonstrates the process of transforming the augmented matrix into this form to solve for the variables x, y, and z.
💡Variables
In the context of a system of linear equations, variables are symbols, usually letters such as x, y, and z, that represent unknown quantities to be solved for. The values of these variables are the solutions to the system of equations. The video focuses on finding the values of the variables x, y, and z by applying Gauss-Jordan Elimination to the given system.
💡Linear Equations
Linear equations are mathematical equations in which the highest power of the variables is 1, and they are typically used to model relationships between different quantities that are proportional to each other. A system of linear equations involves multiple linear equations, and the goal is to find the values of the variables that satisfy all the equations simultaneously. The video is centered around solving such a system using the Gauss-Jordan Elimination method.
💡Pivot
In the context of matrix operations like Gauss-Jordan Elimination, a pivot is an element in a matrix that is used as a multiplier during the row operation process. The pivot is typically the first non-zero element from the left in a row, and it is used to create a leading 1 in the diagonal of the matrix. The process of moving the pivot to the correct position is known as pivoting. In the video, the script describes the operations that lead to creating a pivot in each row and using it to simplify the matrix further.
💡Elementary Row Operations
Elementary row operations are the basic operations that can be performed on a matrix or an augmented matrix during the process of transforming it. These include swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row to another row. These operations are fundamental to the Gauss-Jordan Elimination method and are used to achieve the reduced row echelon form of the matrix.
💡Solving Systems of Equations
Solving systems of equations refers to the process of finding the values of the variables that satisfy all the equations in the system simultaneously. Linear systems are common in mathematics and have various methods for finding their solutions, such as substitution, elimination, and matrix methods like Gauss-Jordan Elimination. The video focuses on using the Gauss-Jordan Elimination method to solve a system of linear equations with three variables.
💡Coefficients
Coefficients are the numerical factors that multiply the variables in a linear equation. They define the relationship between the variables and are essential in forming the system of equations. In the context of the video, the coefficients are used to construct the augmented matrix, which is then manipulated to solve for the variables.
💡Constants
Constants in a system of linear equations are the numerical values found on the right side of the equation that are not variables. They represent the specific numerical values that, when combined with the coefficients and variables, form the equation. In the video, the constants are part of the augmented matrix and are crucial in solving for the values of the variables.
Highlights

Introduction to Gauss-Jordan elimination as a method for solving systems of equations with three variables.

Formulation of the initial augmented matrix from the given system of equations.

The goal of transforming the matrix into reduced row echelon form to find the values of x, y, and z.

Performing elementary row operations to manipulate the matrix, starting with subtracting row1 from row2.

Adjusting the matrix further by adding negative 2 times row1 to row3 to clear the first entry.

Transforming row3 to have a 0 in the second entry by adding two times row3 to row2.

Making the first entry of row1 positive by multiplying row2 by -3 and subtracting it from row1.

Clearing the second entry of row2 by adding row3 to row2.

Correcting a mistake in the calculation and emphasizing the importance of precision in mathematical processes.

Final step of converting the matrix into reduced row echelon form by normalizing the main diagonal entries.

Deriving the solution of the system of equations by interpreting the final form of the matrix.

Assigning the values to variables: x = 6, y = -1, and z = -2 as the solution.

Demonstration of the step-by-step process of Gauss-Jordan elimination, which is both educational and practical.

Explanation of how to perform each row operation and the reasoning behind each step.

Emphasis on the importance of achieving a reduced row echelon form for the matrix to solve the system of equations.

Incorporate the use of visual representation (the matrix) along with the verbal explanation to aid understanding.

Transcripts

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