PreCalculus - Matrices & Matrix Applications (26 of 33) Solving Sys of Linear Eqn with Inverse

Michel van Biezen
15 Jul 201504:04
EducationalLearning
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TLDRThis script offers a clear and concise tutorial on solving systems of linear equations using the inverse matrix method. It explains the process step by step, starting with defining the matrices A, X, and B, and then calculating the inverse of matrix A. By multiplying the inverse of A with matrix B, the script demonstrates how to find the values of x and y that satisfy the given equations. The example provided walks through the calculation in detail, leading to the correct solution of x = -1 and y = 1, which is then verified by substituting the values back into the original equations. The summary emphasizes the practical application of matrix algebra in solving linear systems, highlighting the efficiency and accuracy of the inverse matrix method.

Takeaways
  • 📚 The script provides a method for solving a system of linear equations using the inverse matrix method.
  • 🔢 The example given involves two simple equations: 2x + 5y = 3 and 3x + 3y = 2.
  • 🤔 The key is to identify the matrices A (coefficients), X (variables), and B (constants) from the equations.
  • 🌟 The system of equations can be represented as Ax = B, where A is the matrix of coefficients and B is the matrix of constants.
  • 🎯 To find the solution, calculate the inverse of matrix A (denoted as A^(-1))
  • 📈 The determinant of matrix A is calculated as (2*3) - (5*1) = 6 - 5 = 1.
  • ⚙️ The inverse of matrix A is found by swapping the diagonal elements and changing the signs of the off-diagonal elements: [3 -5; -1 2].
  • 🧠 The solution matrix X is obtained by multiplying the inverse of A with matrix B.
  • 🔍 The values for x and y are found to be -1 and 1, respectively, by performing the matrix multiplication.
  • 📊 Verification is done by plugging the values of x and y back into the original equations to ensure they are satisfied.
  • 📝 The process can be generalized for any system of linear equations by following the same steps: form the matrices, find the inverse of A, and multiply by B.
Q & A

  • What is the main topic of the transcript?

    -The main topic of the transcript is solving a system of linear equations using the inverse matrix method.

  • What are the two simple equations presented in the example?

    -The two simple equations presented are 2x + 5y = 3 and 3x + 3y = 2.

  • What does the matrix A represent in the context of the system of linear equations?

    -In the context of the system of linear equations, matrix A represents the coefficients of the x and y variables.

  • What is the role of the X matrix in this system?

    -The X matrix in this system represents the matrix of the coefficients of the two unknown variables, x and y.

  • What is matrix B in the given example?

    -Matrix B is the matrix that contains the constants from the right side of the equal signs in the system of linear equations, which in this example is [3; 2].

  • How is the inverse of matrix A calculated according to the transcript?

    -The inverse of matrix A is calculated by taking the determinant of A, which is (2*3 - 5*1) = 1, and then swapping the positions of the elements in the matrix and changing the signs of the off-diagonal elements. So, the inverse of A is [-5; -1; 2].

  • What values for x and y satisfy the given system of linear equations?

    -The values for x and y that satisfy the given system of linear equations are x = -1 and y = 1.

  • How can we verify the solution (x, y) in the system of linear equations?

    -We can verify the solution by plugging the values of x and y back into the original equations. For x = -1 and y = 1, 2*(-1) + 5*1 = 3 and 3*(-1) + 3*1 = 2, which confirms that the solution is correct.

  • What is the general process for solving a system of linear equations using the inverse matrix method?

    -The general process involves creating a matrix A from the coefficients of the variables, a matrix X from the variables themselves, and a matrix B from the constants. The system of equations is written as A * X = B. To solve for X, we find the inverse of A and multiply it by B, which gives us the values for x and y.

  • What is the determinant used for in this method?

    -The determinant is used to find the inverse of matrix A, which is essential for solving the system of linear equations using the inverse matrix method.

  • What is the significance of the inverse matrix in solving systems of linear equations?

    -The inverse matrix is significant because it allows us to isolate the variables in the system of linear equations. By multiplying the inverse of matrix A with matrix B, we can directly obtain the values of the unknowns (x and y) that satisfy the system of equations.

Outlines
00:00
📚 Introduction to Solving Systems of Linear Equations with Inverse Matrices

This paragraph introduces the concept of solving systems of linear equations using the inverse matrix method. It presents a simple example involving two equations with variables x and y. The explanation begins by identifying the matrices involved: the 'a' matrix which contains the coefficients of the variables, the 'X' matrix representing the variables themselves, and the 'B' matrix which holds the constants. The paragraph then explains that the solution can be found by setting up the equation 'a * x = B' and solving for 'x' by multiplying the inverse of 'a' with 'B'. The process of finding the inverse of the 'a' matrix is detailed, including calculating the determinant and rearranging the matrix elements. The example concludes with the solution (x = -1, y = 1) and a verification of the solution by plugging the values back into the original equations. The paragraph also provides a quick review of the steps involved in solving such systems of linear equations using matrix inversion.

Mindmap
Keywords
💡System of Linear Equations
A system of linear equations is a set of multiple linear equations that involve the same variables. In the video, the system is represented by two equations, 2x + 5y = 3 and 3x + 3y = 2, which are to be solved simultaneously for values of x and y. The main theme of the video is to demonstrate how to find the solution to such a system using the inverse matrix method.
💡Matrix
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. In the context of the video, matrices are used to represent the system of linear equations in a more compact form, where the coefficients of the variables form matrix A, the variables form matrix X, and the constants form matrix B.
💡Inverse Matrix
The inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. In the context of the video, finding the inverse of matrix A is crucial for solving the system of linear equations, as it allows the calculation of matrix X, which contains the values of x and y satisfying both equations.
💡Determinant
The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix and is used in various ways in mathematics, including finding the inverse of a matrix. In the video, the determinant is calculated for matrix A to facilitate the computation of its inverse.
💡Coefficients
Coefficients are the numerical factors in front of variables in an equation. In the video, the coefficients refer to the numbers multiplying the variables x and y in the system of linear equations.
💡Variables
Variables are symbols, often letters, that represent unknown quantities in an equation. In the video, x and y are the variables for which the viewer is trying to find values that satisfy the given system of linear equations.
💡Constants
Constants are fixed values that do not change. In the context of the video, constants are the numbers on the right side of the equal sign in the system of linear equations.
💡Matrix Multiplication
Matrix multiplication is an operation that takes a pair of matrices, and produces another matrix. In the video, matrix multiplication is used to find the solution to the system of linear equations by multiplying the inverse matrix with the constants matrix.
💡Solution
A solution to a system of linear equations is a set of values for the variables that make both equations true. The video demonstrates how to find this solution using the inverse matrix method.
💡Verification
Verification in the context of solving equations is the process of checking whether the found solution satisfies the original equations. In the video, the solution is verified by substituting the values of x and y back into the equations to ensure they are correct.
💡Equilibrium
In the context of systems of equations, equilibrium refers to the state where all equations are simultaneously satisfied by the same set of values. The video aims to find the equilibrium values for x and y that satisfy both equations in the system.
Highlights

Introduction to solving a system of linear equations using the inverse matrix method.

Two simple equations are given: 2x + 5y = 3 and 3x + 3y = 2.

Matrix A represents the coefficients of x and y variables.

Matrix X represents the matrix of the ex-wives of the two unknown variables (x and y).

Matrix B represents the constants of the right side of the equal signs.

The system of linear equations can be written in matrix form as A * X = B.

To find the values of x and y, we need to calculate the inverse of matrix A and multiply it by matrix B.

Matrix A is defined as the 2x2 matrix [2 5; 1 3].

The determinant of matrix A is calculated as (2*3) - (5*1) = 6 - 5 = 1.

The inverse of matrix A is determined by swapping the elements on the main diagonal and changing the sign of the off-diagonal elements, resulting in the matrix [3 -5; -1 2].

To find the values for x and y, multiply the inverse of matrix A by matrix B [3; 2].

The solution yields x = -1 and y = 1.

Verification of the solution is done by plugging the values of x and y back into the original equations.

The method can be applied to any system of linear equations by creating matrices for the coefficients, variables, and constants.

The system of linear equations can be rewritten as A * X = B, where X is the solution matrix.

Solving for X involves finding the inverse of matrix A and multiplying it by matrix B.

This method provides a systematic approach to finding the solutions of a system of linear equations.

The process is efficient and can be easily understood and implemented.

The use of matrix operations simplifies the process of solving complex systems of equations.

This method has practical applications in various fields, including engineering, economics, and computer science.

Transcripts

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