🔷11 - Gauss Jordan Elimination and Reduced Row Echelon Form

SkanCity Academy

26 May 202215:38

EducationalLearning

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TLDRThis video script outlines the process of solving a system of linear equations using the Gauss-Jordan elimination method. It guides viewers through representing the equations as an augmented matrix and transforming it into reduced row echelon form. The method involves a series of elementary row operations to eliminate variables and isolate the leading diagonal. The script culminates in finding the values of the unknown variables x, y, and z, concluding with x = 1, y = 2, and z = 3.

Takeaways

📌 Gauss-Jordan Elimination is a method used to solve systems of linear equations by transforming an augmented matrix into Reduced Row Echelon Form (RREF).
🔢 The process involves representing the system of equations as an augmented matrix with coefficients of the unknown variables (x, y, z) and constants.
➗ The aim is to zero out elements in the matrix not on the leading diagonal and make the diagonal elements equal to 1 through elementary row operations.
🔄 Elementary row operations include adding or subtracting multiples of one row to another, and multiplying a row by a non-zero scalar.
🎯 The first step is to eliminate the first variable (x) from the second and third equations to create a 'staircase' of zeros below the first pivot.
🔄 Further row operations are performed to eliminate the second variable (y) from the third equation, creating a further staircase of zeros.
📉 When all off-diagonal elements above the pivot are zero, the matrix is ready to be transformed into RREF by making diagonal elements equal to 1.
📈 This is done by dividing each pivot row by the leading diagonal element, normalizing them to 1.
👀 By observing the resulting RREF, we can directly read the values of the variables (x, y, z) from the last column of the matrix.
🏆 The final solution to the system of equations is x = 1, y = 2, and z = 3, as deduced from the last column of the RREF matrix.

Q & A

What method is used in the video to solve the system of linear equations?
-The method used in the video to solve the system of linear equations is the Gauss-Jordan elimination method.
What are the unknown variables the video aims to find?
-The unknown variables the video aims to find are x, y, and z.
How is the system of linear equations represented initially in the Gauss-Jordan elimination method?
-The system of linear equations is represented initially as an augmented matrix, with the coefficients of the unknown variables x, y, z in the rows and the constants on the right-hand side of the equal sign.
What is the goal when transforming the augmented matrix using elementary row operations?
-The goal when transforming the augmented matrix using elementary row operations is to convert it into the reduced row echelon form, where all elements above the leading diagonal are zero and the elements in the leading diagonal are one.
How does the video demonstrate the process of eliminating variables from the system?
-The video demonstrates the process of eliminating variables by performing a series of elementary row operations, such as multiplying rows by constants, adding or subtracting rows from each other, to manipulate the matrix into the desired form.
What specific row operation does the video perform first to start the elimination process?
-The first specific row operation performed in the video is multiplying row three by three and then subtracting that from row two to make the first element of row two zero.
How does the video ensure that the elements in the leading diagonal become one?
-The video ensures that the elements in the leading diagonal become one by dividing each of the rows by the leading diagonal elements, after all other elements above the diagonal have been reduced to zero.
What are the final values of x, y, and z after completing the Gauss-Jordan elimination process?
-The final values of x, y, and z after completing the Gauss-Jordan elimination process are x = 1, y = 2, and z = 3.
What is the significance of the reduced row echelon form in solving systems of linear equations?
-The significance of the reduced row echelon form is that it simplifies the system of linear equations into a form where the solution can be easily read off from the matrix, with each variable corresponding to a pivot position in its respective row.
How does the Gauss-Jordan elimination method differ from other methods of solving systems of linear equations?
-The Gauss-Jordan elimination method differs from other methods as it involves transforming the augmented matrix into the reduced row echelon form, which not only solves the system but also provides a systematic way to deal with inconsistent or dependent systems of equations.

Outlines

00:00

📚 Introduction to Solving Systems of Linear Equations

This paragraph introduces the Gauss-Jordan elimination method for solving a system of linear equations. The goal is to find the unknown variables x, y, and z by representing the equations in the form of an augmented matrix. The initial step involves transforming the matrix into the reduced row echelon form through elementary row operations, aiming to achieve zeros in the non-leading diagonal and ones on the diagonal. The process begins with setting up the matrix according to the given equations and separating constants from the variables.

05:00

🔢 Performing Elementary Row Operations

In this paragraph, the focus is on performing specific elementary row operations to simplify the augmented matrix. The operations include multiplying a row by a certain factor and then adding or subtracting it from another row. This process is used to manipulate the matrix in a way that zeros are created in specific positions. The detailed calculations and the resulting changes in the matrix are explained step by step, showing how the values evolve through each operation.

10:01

🎯 Achieving Zero Values and Finalizing the Matrix

This section describes the continued effort to achieve zero values in specific cells of the matrix through targeted row operations. The process involves a series of multiplications and additions/subtractions, with the aim of reaching a point where the leading diagonal elements are the only non-zero values. The explanation includes the rationale behind each operation and the resulting matrix configuration after each step.

15:01

🏁 Conclusion and Solution of the System of Equations

The final paragraph concludes the Gauss-Jordan elimination process by making the final adjustments to the matrix to find the values of x, y, and z. The operations focus on adjusting the leading diagonal elements to one while maintaining the zeros in other positions. After completing the row operations, the values of the unknown variables are determined by observation. The solution is presented clearly, with z being equal to three, y equal to two, and x equal to one.

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 the reduced row echelon form. The process involves a series of elementary row operations that aim to simplify the matrix, making it easier to read and from which the solutions for the unknown variables can be directly obtained. In the video, this method is used to find the values of x, y, and z for the given system of equations.

💡Linear Equations

Linear equations are mathematical equations in which the highest power of the variable is 1. They represent straight lines in a two-dimensional space and hyperplanes in higher dimensions. In the context of the video, the system of linear equations is the problem that needs to be solved, with the goal of finding the values of the unknown variables x, y, and z that satisfy all the equations simultaneously.

💡Augmented Matrix

An augmented matrix is a matrix that consists of the coefficients of a system of linear equations alongside the constants from the right side of the equations. It is used in methods like Gauss-Jordan Elimination to visually represent and manipulate the system of equations for easier solution. The video uses the augmented matrix to apply row operations and solve for the unknown variables.

💡Elementary Row Operations

Elementary row operations are basic operations performed on rows of a matrix that do not change the solution set of the system of linear equations. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row to another. In the video, elementary row operations are used to transform the augmented matrix into reduced row echelon form.

💡Reduced Row Echelon Form

Reduced row echelon form is a specific way of arranging the rows of a matrix in a simplified, easy-to-read format where the leading diagonal elements are 1, and all other elements in the column are 0. This form is particularly useful for solving systems of linear equations because it makes the solutions clear and straightforward. The video's main goal is to transform the augmented matrix into this form to find the values of the unknown variables.

💡Unknown Variables

Unknown variables are symbols, typically letters like x, y, or z, used in equations to represent values that are not yet known. In the context of the video, the focus is on finding the values of the unknown variables x, y, and z that satisfy the given system of linear equations.

💡Coefficients

Coefficients are numerical factors that multiply variables in an equation. In a linear equation, the coefficients represent the numbers in front of the variables and determine the slope of the line or the direction of the hyperplane. In the video, the coefficients are the numbers that multiply the variables x, y, and z in the system of equations.

💡Constants

Constants are numerical values that do not change and are not variables. In a system of linear equations, constants are the numbers on the right side of the equals sign. They are an essential part of the equations since they, along with the coefficients and variables, determine the solution.

💡Row Operations

Row operations are actions performed on the rows of a matrix or a system of equations. These operations include swapping rows, scaling rows (multiplying by a non-zero scalar), and adding or subtracting one row from another. The purpose of row operations is to manipulate the system to a simpler form, which can help in solving for the unknown variables.

💡Leading Diagonal

The leading diagonal of a square matrix is the diagonal line of cells from the top left corner to the bottom right corner. In the context of the Gauss-Jordan Elimination method, the goal is to have 1's on the leading diagonal and 0's in all other cells of the same columns as the 1's. This makes it easier to read off the solutions of the system of equations.

💡Solving Systems of Equations

Solving systems of equations refers to the process of finding the values of the unknown variables that satisfy all the equations in the system simultaneously. Various methods, including the Gauss-Jordan Elimination method shown in the video, can be used to solve such systems. The solution to a system of equations can be unique, have no solution, or have infinitely many solutions depending on the nature of the system.

Highlights

The video demonstrates solving a system of linear equations using the Gauss-Jordan elimination method.

The goal is to find the unknown variables x, y, and z from the given system of equations.

Linear equations are initially represented in the form of an augmented matrix, with coefficients of the variables and constants.

Elementary row operations are applied to transform the matrix into the reduced row echelon form.

The process involves making elements in the leading diagonal go to one and other elements to zero.

Row three is multiplied by three and subtracted from row two to eliminate the first element in row three.

Row three is further manipulated to make the element in the second position zero by multiplying and subtracting from row one.

Row two is modified by multiplying it by negative five to make the element in the second position match the leading diagonal element.

Row three is adjusted to have zero in the second position by multiplying row two by negative five and adding it to row three.

Row one is modified by multiplying row three by five and adding it to row one to make the second element zero.

Row two is adjusted by multiplying row three by two and adding it to row two to make the second element zero.

Row one is modified by multiplying row two by negative three and adding it to row one to make the second element zero.

The leading diagonal elements are normalized to one by dividing each row by its leading element.

The final values of x, y, and z are determined to be 1, 2, and 3, respectively.

The method provides a systematic approach to solving systems of linear equations with three variables.

Gauss-Jordan elimination is an effective technique for finding exact solutions to linear systems.

The process requires careful execution of row operations to maintain the integrity of the solution path.

Transcripts

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