PreCalculus - Matrices & Matrix Applications (28 of 33) Solving Sys of Linear Eqn with Inverse

Michel van Biezen
17 Jul 201509:05
EducationalLearning
32 Likes 10 Comments

TLDRThis video script introduces a method for solving a 3x3 system of linear equations using matrix inversion. It explains the process of finding the variables x, y, and z by using a matrix A with coefficients from the equations, matrix X for the variables, and matrix B for the constants. The key step involves finding the inverse of matrix A by using the Gauss-Jordan elimination method and an augmented matrix with an identity matrix. The script provides a detailed walkthrough of the row operations required to transform the matrix into its inverse, ultimately allowing the calculation of the solution for the variables x, y, and z.

Takeaways
  • 📐 The lesson focuses on expanding the technique of solving linear equations to a 3x3 system, involving variables x, y, and z.
  • 🔍 The concept involves using a matrix A that contains coefficients from the three equations, Matrix X for variables, and Matrix B for constants.
  • 🤖 The solution is obtained by finding the inverse of Matrix A and multiplying it by Matrix B, which yields the values of x, y, and z.
  • 🧩 The Gauss-Jordan method of elimination is used to manipulate the augmented matrix, which combines Matrix A with an identity matrix on its right.
  • 🔄 The goal of the Gauss-Jordan method is to turn the left side of the augmented matrix into the identity matrix, revealing the inverse of Matrix A on the right side.
  • 🌀 The process starts by ensuring there is a '1' in the top left corner of the matrix, achieved by swapping rows if necessary.
  • 🔢 Each step involves using row operations to create zeros below the '1' in each column, working from left to right.
  • ✅ Once the left side of the augmented matrix becomes the identity matrix, the right side represents the inverse of Matrix A.
  • 📈 The values of x, y, and z are found by multiplying the inverse of Matrix A with Matrix B.
  • 🔍 The script provides a detailed example, walking through the steps of the Gauss-Jordan method and the resulting values of the variables.
  • 📝 The method demonstrated is a systematic approach to solving systems of linear equations involving three variables.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is the process of solving a system of linear equations using a 3x3 matrix and the Gauss-Jordan elimination method.

  • What are the three matrices involved in this process?

    -The three matrices involved are Matrix A (with coefficients of X, Y, and Z), Matrix X (representing the variables X, Y, and Z), and Matrix B (representing the constants from the right side of the equations).

  • How is the inverse of Matrix A found?

    -The inverse of Matrix A is found by creating an augmented matrix with Matrix A and an identity matrix, and then using the Gauss-Jordan elimination method to transform the left side into the identity matrix. The right side of the matrix then represents the inverse of Matrix A.

  • What is the Gauss-Jordan elimination method?

    -The Gauss-Jordan elimination method is a technique used to solve systems of linear equations by transforming the augmented matrix's left side into the identity matrix through a series of row operations, resulting in the inverse matrix on the right side.

  • What row operations are performed to achieve the identity matrix on the left side?

    -The row operations performed include row exchanges, multiplying a row by a scalar, and adding or subtracting multiples of one row to another row to achieve the identity matrix on the left side of the augmented matrix.

  • How are the variables X, Y, and Z found?

    -Once the inverse of Matrix A is found, the variables X, Y, and Z are calculated by multiplying the inverse matrix by Matrix B, which contains the constants from the equations.

  • What is the resulting values for X, Y, and Z in the given example?

    -In the given example, the resulting values are X = -60, Y = -34, and Z = -5.

  • How can you verify the solution is correct?

    -The solution can be verified by substituting the values of X, Y, and Z back into the original equations and checking if both sides of the equations are equal.

  • Why is it important to have a one in the diagonal positions when using the Gauss-Jordan method?

    -Having a one in the diagonal positions is important because it allows for the elimination of variables from other rows and columns through row operations, which is essential for the Gauss-Jordan elimination method to work effectively.

  • What is the significance of the identity matrix in the augmented matrix?

    -The identity matrix in the augmented matrix serves as a placeholder for the inverse of Matrix A. Once the left side is transformed into the identity matrix, the right side (originally the identity matrix) now holds the values of the inverse, which can be used to solve for the variables.

  • How does this technique differ from other methods of solving systems of linear equations?

    -This technique differs from other methods such as substitution or elimination because it uses matrix operations to solve the system, which can be more efficient and applicable to larger systems of equations. It also allows for the direct computation of the inverse matrix, which can be useful in various mathematical and computational contexts.

Outlines
00:00
🧠 Solving 3x3 Systems of Linear Equations Using Matrices

The video begins with an introduction to solving 3x3 systems of linear equations by utilizing matrices. The presenter explains the setup involving matrix A (coefficient matrix), matrix X (variables x, y, z), and matrix B (constants). The solution involves finding the inverse of matrix A using the Gauss-Jordan elimination method, described step-by-step. The process starts by adjusting rows for strategic zeroes and ones in matrix A, facilitating the conversion into an identity matrix, while the augmented part transforms into the inverse of matrix A.

05:01
🔍 Detailed Steps in Matrix Manipulation and Solution Calculation

The second paragraph delves deeper into the matrix manipulation steps, illustrating the final stages of using the Gauss-Jordan method to achieve the identity matrix on the left side of the augmented matrix, revealing matrix A's inverse on the right side. It then details how to use this inverse to multiply with matrix B to determine the variables x, y, and z. Each step of the matrix calculation is articulated, leading to the final values of the variables, which are verified to ensure their correctness in the context of the original equations.

Mindmap
Keywords
💡3X3 system
A 3X3 system refers to a set of three linear equations with three variables, typically denoted as x, y, and z. In the context of the video, the system is used to demonstrate the process of solving for the variables by employing matrix operations. The script mentions expanding the technique to a 3X3 system, indicating a progression from simpler systems to more complex ones, and showcasing the versatility of the method being discussed.
💡Matrix A
Matrix A is a square matrix that contains the coefficients of the variables x, y, and z from the three linear equations. It is a fundamental component in the process of solving the system, as the inverse of Matrix A is used to find the values of the variables. The video script describes how to manipulate Matrix A, along with the identity matrix, to find its inverse and ultimately solve for the variables.
💡Matrix X
Matrix X represents the variables x, y, and z in the system of linear equations. It is used in conjunction with Matrix A and Matrix B to solve for the values of the variables. In the video, Matrix X is mentioned as the matrix that will be used to find the values of x, y, and z once the inverse of Matrix A is calculated and multiplied by Matrix B.
💡Matrix B
Matrix B is a column matrix that contains the constants from the right side of the equal sign in the system of linear equations. It works in tandem with Matrix A and Matrix X to form the augmented matrix, which is then manipulated to find the solution to the system. In the context of the video, Matrix B is essential for the final step of solving the equations, as it is multiplied by the inverse of Matrix A to find the values of x, y, and z.
💡Inverse of Matrix A
The inverse of Matrix A is a matrix that, when multiplied by Matrix A, results in the identity matrix. It is a crucial concept in the video, as finding the inverse allows for the solution of the system of linear equations. The script describes the process of using the Gauss-Jordan method of elimination to find the inverse, which is then used to determine the values of the variables x, y, and z.
💡Gauss-Jordan method
The Gauss-Jordan method is an elimination technique used to find the inverse of a matrix and solve systems of linear equations. It involves a series of row operations that transform the augmented matrix into a form where the left side is the identity matrix and the right side is the inverse of the original matrix. The video script provides a step-by-step walkthrough of this method, illustrating how it is applied to the 3X3 system to find the inverse of Matrix A and solve for the variables x, y, and z.
💡Row operations
Row operations are a set of basic operations performed on rows of a matrix during the process of matrix manipulation, such as swapping rows, multiplying a row by a scalar, or adding/subtracting one row from another. These operations are fundamental to the Gauss-Jordan method described in the video, which uses them to transform the augmented matrix into a form that allows for the calculation of the inverse and the solution of the system of equations.
💡Identity Matrix
An identity matrix is a special square matrix with ones on the diagonal and zeros elsewhere. It is used as a fundamental structure in linear algebra, and when multiplied by another matrix, it leaves the other matrix unchanged. In the video, the identity matrix is part of the augmented matrix, and the goal of the Gauss-Jordan method is to transform the left side of the augmented matrix into the identity matrix, which simultaneously reveals the inverse of the original matrix on the right side.
💡Augmented Matrix
An augmented matrix is a combination of a coefficient matrix and a constant matrix placed side by side. It is used in systems of linear equations to facilitate the process of finding the inverse of the coefficient matrix and solving for the variables. In the video, the augmented matrix is created by attaching the identity matrix to the right of Matrix A, and through a series of row operations, it is transformed to separate the identity matrix on the left from the inverse of Matrix A on the right.
💡Linear equations
Linear equations are mathematical equations in which the highest power of the variables is one, and they represent straight lines in two-dimensional space. The video is focused on solving a system of linear equations using matrix operations, specifically by finding the inverse of a matrix and using it to determine the values of the variables that satisfy all equations in the system.
💡Variables
In the context of the video, variables refer to the unknowns, denoted as x, y, and z, that are being solved for in the system of linear equations. The process described in the video aims to find the values of these variables that satisfy all three equations simultaneously, using matrix operations and the Gauss-Jordan method.
Highlights

Introduction to expanding techniques to a 3x3 system of linear equations.

Explanation of the matrix components: Matrix A with coefficients, Matrix X with variables, and Matrix B with constants.

The concept of solving for variables x, y, z using the inverse of Matrix A multiplied by Matrix B.

Introduction to the use of the Gauss-Jordan method for finding the inverse of Matrix A.

Steps to augment Matrix A with the identity matrix to prepare for manipulation.

Detailed process of row exchanges to position a leading one in the matrix.

Use of elementary row operations to transform Matrix A into the identity matrix.

The method to turn specific matrix elements to zero using controlled row operations.

Explanation and execution of row swapping to manipulate the matrix further.

Continued application of the Gauss-Jordan method to achieve row echelon form.

Completion of the identity matrix on the left side indicating the inverse matrix on the right.

The inverse of Matrix A achieved and verified through transformation.

Calculation of variables x, y, z by multiplying the inverse matrix with Matrix B.

Step-by-step calculation of each variable using matrix multiplication.

Verification of the solution, confirming the correctness of the calculated values for x, y, z.

Summary of the process, emphasizing the use of matrices and the Gauss-Jordan elimination method in solving systems of linear equations.

Transcripts

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LinearAlgebraGaussJordanMatrixInversionEquationSolvingMathematicsEducationalContentStepByStepVariableCalculationMathTutorialsLinearSystems