PreCalculus - Matrices & Matrix Applications (7 of 33) Method of Gaussian Elimination: 3x3 Matrix*
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
📚 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.
🔢 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
💡System of Linear Equations
💡Gaussian Elimination
💡Three by Three Augmented Matrix
💡Diagonal Elements
💡Row Operations
💡Back-Substitution
💡Constants
💡Coefficients
💡Variables
💡Solution Set
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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