Solving System of Linear Equations: Gaussian Elimination

Nadine Alex Bravo
22 May 202019:58
EducationalLearning
32 Likes 10 Comments

TLDRThis video script offers a comprehensive guide on solving systems of linear equations using the Gaussian elimination method. It explains the creation of an Augmented matrix and the process of transforming it into a row echelon form through elementary row operations. The script details each step, including the calculation of leading entries and the use of formulas to achieve a zero entry in specific positions. Ultimately, it demonstrates how to arrive at the solution by systematically following the Gaussian elimination process, highlighting its effectiveness in finding the values of the variables involved.

Takeaways
  • 📚 The Gaussian elimination method is a technique for solving systems of linear equations by transforming the augmented matrix into row echelon form.
  • 🔢 An augmented matrix is created by writing the coefficient matrix followed by a colon and the constants from each equation.
  • 📈 Row echelon form is characterized by having leading entries (non-zero values) in the diagonal, with zeros below each leading entry.
  • 🔄 The process involves performing a series of elementary row operations to manipulate the matrix into the desired form.
  • 🌟 Two types of echelon forms exist: row echelon form and reduced row echelon form, with the latter being more simplified.
  • 🤝 The leading entries in the matrix should be to the right of the previous leading entry in the row above it.
  • 🧮 To achieve row echelon form, certain rows may need to be multiplied or divided by constants to create leading entries.
  • 🔢 Elementary row operations include adding or subtracting multiples of one row to another to manipulate the entries.
  • 📊 The goal of Gaussian elimination is to transform the coefficient matrix into an upper triangular form, which simplifies the process of finding the solutions.
  • 💡 Once in row echelon form, back substitution can be used to solve for the variables starting with the lowest variable.
  • 📝 The method can be applied to various types of systems, and it is a fundamental technique in linear algebra for solving such equations.
Q & A

  • What is the Gaussian elimination method used for?

    -The Gaussian elimination method is used for solving systems of linear equations by transforming the coefficient matrix into an upper triangular form or, ultimately, reducing it to a row echelon form or a reduced row echelon form.

  • What is an Augmented matrix?

    -An Augmented matrix is a matrix that combines the coefficient matrix of a system of linear equations with its constant terms, separated by a colon after each row of coefficients.

  • What are the two types of echelon forms?

    -The two types of echelon forms are row echelon form and reduced row echelon form. The row echelon form has nonzero entries only at the diagonal and above, while the reduced row echelon form has a leading 1 in each row, with zeros above and below the diagonal entries.

  • What is the goal of transforming the Augmented matrix in Gaussian elimination?

    -The goal of transforming the Augmented matrix in Gaussian elimination is to achieve a row echelon form, where the leading entries (coefficients) are aligned in a diagonal pattern, with zeros below each leading entry.

  • What are elementary row operations?

    -Elementary row operations are basic operations performed on rows of a matrix that preserve the solution set of the system of equations. They include swapping rows, multiplying a row by a nonzero constant, and adding or subtracting a multiple of one row to another row.

  • How do you perform an elementary row operation to make the first entry in the Augmented matrix zero?

    -To make the first entry in the Augmented matrix zero, you can use the operation of subtracting a multiple of the first row from the other rows, ensuring that the first entry becomes zero without affecting the solution set of the system.

  • What is the purpose of achieving an upper triangular matrix in Gaussian elimination?

    -Achieving an upper triangular matrix in Gaussian elimination simplifies the system of equations, making it easier to solve by back substitution. It allows for the direct determination of the values of the variables from the top down.

  • How do you find the solution to a system of linear equations once you have an upper triangular matrix?

    -Once you have an upper triangular matrix, you can find the solution by performing back substitution. You start with the last equation and solve for the last variable, then substitute this value into the equation above it and solve for the next variable, and so on, until you have found all the variable values.

  • What is the significance of the least common multiple (LCM) in the process of transforming the Augmented matrix?

    -The least common multiple (LCM) is used to ensure that the leading entries in the matrix are multiples of each other, which helps in aligning the diagonal pattern required for the row echelon form. It guides the multiplication of rows to achieve the necessary alignment of leading entries.

  • How does the script demonstrate the equivalence of different matrix solution methods?

    -The script demonstrates the equivalence of different matrix solution methods by showing that the same answers can be obtained using Gaussian elimination as were obtained using Cramer's rule and matrix inversion. This shows that different methods can lead to the same solution for a system of linear equations.

  • What is the final result of the example provided in the script?

    -The final result of the example provided in the script is the solution to the system of linear equations, which is x = 49/11, y = -20/11, and z = -21/11. These values satisfy the system when substituted back into the original equations.

Outlines
00:00
📚 Introduction to Gaussian Elimination

This paragraph introduces the Gaussian elimination method for solving systems of linear equations. It explains the initial step of creating an augmented matrix using the coefficient matrix and constant terms. The goal is to transform this augmented matrix into a row echelon form through a series of elementary row operations. The explanation includes the definition of row echelon form and the importance of leading entries, which are the first nonzero entries in each row. The process of transforming the matrix is outlined, emphasizing the need for leading entries to be positioned to the right of the previous leading entry, ultimately resulting in an upper triangular form.

05:01
🔄 Performing Elementary Row Operations

This paragraph delves into the specifics of performing elementary row operations to achieve a row echelon form. It describes how to manipulate the matrix rows to create a target element of zero in specific positions. The process involves using different rows to create new equations and adjust the elements accordingly. The explanation includes the use of subtraction and multiplication to achieve the desired row form. The paragraph also highlights the importance of the least common multiple when dealing with different leading coefficients and the application of formulas to transform the matrix effectively.

10:04
📈 Achieving Row Echelon and Upper Triangular Forms

This paragraph continues the discussion on transforming the augmented matrix into row echelon and upper triangular forms. It explains how to apply formulas to each element in a row to achieve a zero target and how to use the least common multiple to simplify the process. The paragraph details the step-by-step process of achieving the desired matrix form, including the use of subtraction and multiplication of rows. It also discusses the final steps to arrive at the solution of the system of equations, emphasizing the use of back substitution to find the values of the variables.

15:06
🎓 Solving the System: Cramer's Rule and Matrix Inversion

The final paragraph concludes the process of solving the system of linear equations using Gaussian elimination. It describes how to solve for the variables by applying the formulas derived from the transformed matrix. The paragraph explains the use of back substitution to find the values of Y and X, and how these values can be cross-checked using Cramer's rule and matrix inversion methods. The summary emphasizes that the same answers can be obtained through different methods, reinforcing the understanding of the Gaussian elimination process and its effectiveness in solving systems of linear equations.

Mindmap
Keywords
💡Gaussian Elimination
Gaussian Elimination is a method used to solve systems of linear equations. It involves transforming an augmented matrix into an upper triangular matrix through a series of elementary row operations. In the video, the process is explained step by step, showing how to achieve a row echelon form and eventually solve for the variables.
💡Augmented Matrix
An Augmented Matrix is a matrix that combines the coefficient matrix and the constant matrix of a system of linear equations. It is used in Gaussian Elimination to keep track of both the coefficients of the variables and the constants in each equation. The script mentions creating an augmented matrix and transforming it throughout the process.
💡Row Echelon Form
Row Echelon Form is a particular way of arranging the rows of a matrix in a simplified structure where each row has a leading coefficient (also known as a pivot) that is to the right of the pivot of the row above it. This form is an intermediate step in the Gaussian Elimination process, which helps in solving the system of equations.
💡Elementary Row Operations
Elementary Row Operations are basic operations performed on rows of a matrix during the Gaussian Elimination process. These include adding or subtracting multiples of one row to another, swapping rows, and multiplying a row by a nonzero constant. These operations help transform the matrix into the desired form without changing the solution to the system of equations.
💡Leading Coefficient
The Leading Coefficient, or Pivot, is the first nonzero coefficient in a row of the matrix from the left. In the context of Gaussian Elimination, the goal is to create a row echelon form where the leading coefficients are aligned in a staircase pattern down the matrix.
💡Upper Triangular Matrix
An Upper Triangular Matrix is a square matrix in which all the entries below the main diagonal are zero. This type of matrix is the final form achieved after applying Gaussian Elimination, which makes it easier to solve for the variables in the system of equations.
💡Systems of Linear Equations
A System of Linear Equations is a set of equations where each equation is linear and has the same variables. These systems can be solved using various methods, including Gaussian Elimination, to find the values of the variables that satisfy all equations simultaneously.
💡Coefficient Matrix
The Coefficient Matrix is a square matrix that contains the coefficients of the variables in a system of linear equations. It is one part of the augmented matrix used in solving the system through Gaussian Elimination.
💡Constant Matrix
The Constant Matrix is a matrix that contains the constants or the values on the right-hand side of the equations in a system of linear equations. It is paired with the coefficient matrix to form the augmented matrix for solving the system using Gaussian Elimination.
💡Back Substitution
Back Substitution is a technique used in solving systems of linear equations, particularly after the matrix has been transformed into an upper triangular form. It involves solving for the variables starting from the last equation and working backward, substituting the values of the variables into the previous equations.
Highlights

Introduction to solving systems of linear equations using the Gaussian elimination method.

Explanation of creating an Augmented matrix, which is essential for the Gaussian elimination process.

Description of the row echelon form, a key intermediate matrix form in Gaussian elimination.

Details on the two types of echelon forms: row echelon form and reduced row echelon form.

Importance of leading entries and how they are used to transform the Augmented matrix.

Step-by-step process of performing elementary row operations to achieve a row echelon form.

Demonstration of how to manipulate the matrix to get a specific target element to zero.

The requirement for the leading entries to form an upper triangular pattern in the matrix.

Explanation of how to use the first row to transform other rows in the matrix.

Procedure for making the first element of the third row a non-zero value by finding the least common multiple.

Transformation of the Augmented matrix into an upper triangular matrix through a series of row operations.

Solving for the variables in the system of equations using back substitution after achieving the upper triangular matrix.

Comparison of Gaussian elimination with other methods like Cramer's rule and matrix inversion, showing their equivalence in finding the solution.

Final solution of the system of linear equations using Gaussian elimination method.

Transcripts

Browse More Related Video

Gaussian Elimination & Row Echelon Form

PreCalculus - Matrices & Matrix Applications (5 of 33) Method of Gaussian Elimination: Example

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

Gaussian Elimination With 4 Variables Using Elementary Row Operations With Matrices

A Guide to Gaussian Elimination Method (and Solving Systems of Equations) | Linear Algebra

Ch. 10.3 Matrices and Systems of Linear Equations

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
LinearAlgebraGaussianEliminationAugmentedMatrixRowEchelonFormUpperTriangularMathTutorialsEducationalContentProblemSolvingMathematicsLinearSystems