PreCalculus - Matrices & Matrix Applications (7 of 33) Method of Gaussian Elimination: 3x3 Matrix*

Michel van Biezen
10 Jun 201506:39
EducationalLearning
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TLDRThe video script presents a step-by-step guide on solving a three-variable linear equation system using Gaussian elimination. It explains the process of transforming the system into an augmented matrix and then systematically manipulates the matrix to have ones on the diagonal and zeros elsewhere. The method involves row operations, including swapping and scaling, to simplify the system and ultimately find the values of x, y, and z. The script concludes with a verification of the solution, demonstrating the effectiveness of Gaussian elimination in solving complex systems of linear equations.

Takeaways
  • 📚 The script introduces a method for solving a three-equation, three-unknown problem, also known as a system of linear equations.
  • 🔢 The Gaussian elimination method is used, which involves manipulating a three-by-three augmented matrix.
  • 🎯 The goal is to transform the matrix into a form where all diagonal elements are ones and all off-diagonal elements are zeros.
  • 🔄 The process starts with the top-left element, which is already one, and no changes are needed for the first row.
  • ➗ To eliminate unwanted numbers, rows are replaced by modified versions (multiplied and added) of other rows with key diagonal elements.
  • 🔄 After the initial elimination, the second row is modified by dividing by the key diagonal element (-2) to get a positive one.
  • 🔄 Further row operations are performed to get zeros in the correct positions, such as replacing rows with multiples of other rows.
  • 📈 The final steps involve normalizing rows to get ones on the diagonal and eliminating the last non-zero elements below the diagonal.
  • 🎉 The solution is found when the matrix has ones on the diagonal and zeros elsewhere, representing the values of x, y, and z.
  • 💡 The solution set is x = -1, y = 0, and z = 1, which is verified by substituting these values back into the original equations.
  • 📝 Gaussian elimination is an efficient method for solving systems of linear equations and is well-demonstrated in the script.
Q & A
  • What type of problem is being solved in the transcript?

    -A three equation three unknown type of problem, also known as a system of linear equations, is being solved.

  • Which method is used to solve the system of linear equations in the transcript?

    -The method of Gaussian elimination is used, specifically with a three by three augmented matrix.

  • What is the initial augmented matrix like in the problem?

    -The initial augmented matrix has ones on the diagonal, with the coefficients of the variables (x, y, z) on the left and constants on the right side of the equal sign.

  • How does the process of Gaussian elimination transform the matrix?

    -Gaussian elimination transforms the matrix by using row operations to create a diagonal of ones and zeros elsewhere, ultimately leading to a solution for the variables.

  • What is the first step in the Gaussian elimination process described in the transcript?

    -The first step is to eliminate the numbers below the top left element (which is already one), by replacing the second and third rows with modified versions of themselves through row operations.

  • What row operation is performed to make the second element on the diagonal become one?

    -The second row is divided by its leading coefficient (negative two), turning the -1 into a positive 1.

  • How is the third element on the diagonal made one?

    -The third row is divided by its leading coefficient (negative four), turning the 0 into a 1.

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

    -The final solution set is x equals negative one, y equals zero, and z equals positive one.

  • How is the solution verified in the transcript?

    -The solution is verified by plugging the values of x, y, and z back into the original equations to ensure the left side equals the right side.

  • What is the significance of the method of Gaussian elimination in solving systems of linear equations?

    -The method of Gaussian elimination is significant as it provides a systematic and efficient way to solve systems of linear equations, which is crucial in various fields of mathematics, science, and engineering.

Outlines
00:00
📚 Introduction to Solving a System of Linear Equations

The paragraph begins with an introduction to solving a three-variable linear equation problem, commonly known as a system of linear equations. The method of Gaussian elimination is chosen for this task, utilizing a 3x3 augmented matrix. The initial augmented matrix is constructed with coefficients of the x, y, and z variables on the left and constants on the right. The goal is to manipulate the matrix into a form where the diagonal elements are ones and all other elements are zeros. The process starts with the top left element, which is already a one, and proceeds to eliminate the numbers below it by using a series of row operations. The first row is left unchanged, while the second and third rows are adjusted to achieve the desired matrix form.

05:00
🔢 Completing the Gaussian Elimination Process

This paragraph details the completion of the Gaussian elimination process. It describes the steps taken to transform the matrix into a form where the diagonal elements are ones and the off-diagonal elements are zeros. The process involves a series of row operations, including replacing rows with modified versions to achieve the desired outcomes. The second row is adjusted to have a positive one, and the third row is manipulated to have zeros in the first two positions. The final step involves making the last diagonal element a one by dividing the third row by its diagonal value. The resulting matrix has the solution for the system of linear equations, with x equals negative one, y equals zero, and z equals positive one. A quick check confirms the correctness of the solution.

Mindmap
Keywords
💡Electron Line
The term 'Electron Line' appears to be the name of the series or the show from which the transcript is taken. It is likely a program or platform focused on solving mathematical problems or providing educational content. In the context of the video, it serves as the introductory phrase to set the scene for the viewer, indicating that they are about to learn about solving a complex mathematical problem.
💡System of Linear Equations
A system of linear equations refers to a set of three or more linear equations that involve the same set of variables. Each equation in the system is a straight line, and the goal is to find the point(s) of intersection of these lines in the coordinate plane, which represent the solution(s) to the system. In the video, the focus is on solving a system involving three unknowns and three equations, which is a specific case of a larger category of problems.
💡Gaussian Elimination
Gaussian elimination is a method used to solve systems of linear equations by transforming the system's matrix into an upper triangular form or reduced row echelon form through a series of row operations. The process simplifies the system, making it easier to find the solution by back-substitution. In the context of the video, Gaussian elimination is the chosen method for solving the given system of equations.
💡Three by Three Augmented Matrix
An augmented matrix is a matrix that combines the coefficient matrix of a system of linear equations with its constants in a separate column. In the context of the video, a 'three by three augmented matrix' refers to a matrix that has three rows and three columns for the coefficients, plus an additional column for the constants from the right-hand side of the equations. This matrix is used as a visual tool in the Gaussian elimination method to solve the system of equations.
💡Diagonal Elements
In a square matrix, the diagonal elements are the elements located at the positions where the row index and column index are the same, typically from the top left to the bottom right of the matrix. In the context of Gaussian elimination, the goal is often to manipulate the matrix in such a way that the diagonal elements become ones (1s), and the off-diagonal elements become zeros (0s), which simplifies the process of finding the solution to the system of equations.
💡Row Operations
Row operations are the basic steps used in Gaussian elimination to transform the matrix. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another. The purpose of these operations is to create a matrix that is easier to solve by getting the desired form for the system of equations.
💡Back-Substitution
Back-substitution is the final step in solving a system of linear equations using Gaussian elimination. After the matrix has been transformed into an upper triangular form or reduced row echelon form, the solution is found by substituting values from the last equation into the previous ones, working backwards up the matrix.
💡Constants
In the context of a system of linear equations, constants are the numerical values that are not variables and are placed on the right side of the equal sign in each equation. They represent the numbers that, when combined with the variables and their coefficients, make the equation true.
💡Coefficients
Coefficients are the numerical factors that multiply the variables in a linear equation. They determine the slope of the line represented by the equation in the coordinate plane. In the context of the video, the coefficients are the numbers in front of the variables x, y, and z in the system of equations.
💡Variables
Variables are symbols, often letters like x, y, or z, that represent unknown quantities in a mathematical equation. In a system of linear equations, variables are what the solver seeks to find the values for, to satisfy all equations simultaneously.
💡Solution Set
A solution set is the set of all values for the variables that satisfy all the equations in a system of linear equations simultaneously. In the context of the video, the solution set is the combination of values for x, y, and z that make all three given equations true.
Highlights

Introduction to solving a three equation three unknown problem, also known as a system of linear equations.

Use of Gaussian elimination method with a three by three augmented matrix for problem-solving.

Transformation of the augmented matrix with coefficients of x, y, and z variables and constants on the right side.

Objective to turn diagonal elements into ones and other elements into zeros using row operations.

Elimination of numbers in the second row by replacing it with negative multiples of the top row and adding to the second row.

Resulting matrix after the elimination process, with the first row unchanged and second row transformed.

Transformation of the second row into a positive one by dividing the whole row by negative two.

Further row operations to turn certain elements into zeros, using negative multiples and additions.

Adjustment of the third row to turn its diagonal element into one by dividing the whole row by a negative four.

Final step of turning the off-diagonal elements into zeros by adding negative multiples of the third row to the first and second rows.

Achievement of a solution with ones across the diagonal and zeros elsewhere, representing the values of x, y, and z.

Solution set for the system of linear equations is x equals negative one, y equals zero, and z equals positive one.

Quick check to validate the solution by substituting the values of x, y, and z back into the original equations.

Confirmation of the proper solution through verification checks, demonstrating the effectiveness of Gaussian elimination method.

Overview of the Gaussian elimination method as a slick and efficient way to solve three-equation linear problems.

Transcripts
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