How to find The Inverse of a (3 Γ— 3) matrix Using Determinant and Co-factor

Our Online Tutor
13 Feb 202413:48
EducationalLearning
32 Likes 10 Comments

TLDRThis video tutorial explores the method of finding the inverse of a 3x2 matrix using the cofactor (co-actor) approach. The instructor begins by defining Matrix A and systematically calculates the cofactors for each element, adjusting for sign alternation across the matrix. The determinant of the submatrices is found, and these cofactors are used to create the adjugate matrix by transposing the cofactor matrix. The determinant of Matrix A is then calculated, and using this, the inverse of Matrix A is derived. The tutorial concludes with a practice exercise for the viewers, encouraging them to apply the cofactor method to another matrix and share their results.

Takeaways
  • πŸ“Œ The script focuses on finding the inverse of a 3x2 matrix using the cofactor approach.
  • πŸ”’ Matrix A is given with elements: [1 2; 1 -1; 2 1].
  • 🌟 The cofactor matrix will have dimensions similar to the original matrix, and its elements are signed according to a specific pattern (positive, negative, positive, etc.).
  • πŸ“ The first element of the cofactor matrix is calculated by eliminating the first row and column of Matrix A and finding the determinant of the remaining submatrix.
  • πŸ” Each element of the cofactor matrix is determined by following the pattern of eliminating the appropriate row and column from Matrix A and calculating the determinant.
  • βœ… The determinant of a submatrix is calculated by eliminating one row and one column and multiplying the remaining elements.
  • πŸŒ€ The sign of the determinant alternates, following the pattern of (-1)^(n+m) where n is the row number and m is the column number.
  • πŸ“ The adjugate matrix (also called the adjoint matrix) of A is found by taking the transpose of the cofactor matrix.
  • πŸ”’ The inverse of Matrix A is found by dividing 1 by the determinant of A and multiplying it by the adjugate matrix.
  • πŸ“Š An exercise is provided for the viewer to find the inverse of another matrix B with elements: [2 -3; 5 1; 6 0].
  • πŸ’‘ The video aims to help viewers understand the process of finding the inverse of a matrix using the cofactor approach and encourages them to practice with the provided exercise.
Q & A
  • What is the main topic of the video script?

    -The main topic of the video script is finding the inverse of a 3x2 matrix using the cofactor approach.

  • What is the first step in finding the inverse of a matrix?

    -The first step in finding the inverse of a matrix is to solve for the cofactors of the matrix.

  • How does the sign pattern work for the cofactors in a 3x3 matrix?

    -The sign pattern for the cofactors in a 3x3 matrix alternates as negative, positive, negative, positive, negative, positive.

  • What is the determinant of the 2x2 matrix formed by the first column of the given matrix?

    -The determinant of the 2x2 matrix formed by the first column of the given matrix is -1 * (-2) - 1 * 2, which equals 0.

  • How is the cofactor matrix transposed to find the adjugate matrix?

    -The adjugate matrix is found by transposing the cofactor matrix, which involves swapping the columns with the corresponding rows.

  • What is the determinant of the given matrix A?

    -The determinant of the given matrix A is calculated by multiplying the elements on the main diagonal and subtracting the product of the elements on the other diagonal, resulting in 15.

  • How is the inverse of matrix A found?

    -The inverse of matrix A is found by taking the adjugate matrix, dividing each element by the determinant of A, and then multiplying by the reciprocal of the determinant.

  • What is the final form of the inverse of matrix A?

    -The final form of the inverse of matrix A is a 3x3 matrix with elements 0, 1/3, 1/5; 1/5, 2/3, 1/5; and 1/15, 0, 1/3.

  • What is the exercise given at the end of the video script?

    -The exercise given at the end of the video script is to find the inverse of a given matrix B with elements 2, -3, 5; 6, 0, 1; and 15, -7, 2.

  • How can viewers share their solutions to the exercise?

    -Viewers are encouraged to share their solutions to the exercise in the comment section of the video.

  • What is the significance of the cofactor approach in matrix operations?

    -The cofactor approach is significant in matrix operations as it provides a method to calculate the inverse of a matrix, which is crucial for various mathematical and engineering applications.

Outlines
00:00
πŸ“Œ Introduction to Finding the Inverse of a 3x2 Matrix

This paragraph introduces the process of finding the inverse of a 3x2 matrix using the Cauchy matrix approach. It begins by defining the given matrix A with elements 1, 2, 1, 2, -1, 2, 1, 1, -2 and explains the goal of finding its inverse, denoted as A^-1. The method involves solving for the cofactors of matrix A, emphasizing the importance of their signs which alternate in a specific pattern. The paragraph outlines the steps to calculate the cofactors by eliminating certain elements from the original matrix and computing the determinants of the resulting submatrices. It also highlights the dimension of the matrix and its relation to the dimensions of the cofactors.

05:00
πŸ”’ Detailed Steps for Calculating Cofactors and Inverse

This paragraph delves into the detailed steps of calculating the cofactors and the inverse of matrix A. It explains how to eliminate elements from the original matrix to find the determinants that form the cofactors. The paragraph provides a step-by-step guide on how to compute the determinants for each cofactor, including the handling of signs and the resulting values. It then describes how to transpose the cofactors to find the adjugate matrix of A, which is essential for calculating the inverse. The paragraph concludes with the presentation of the final adjugate matrix and the method to find the inverse by dividing the adjugate matrix by the determinant of the original matrix.

10:02
πŸ“ Exercise: Finding the Inverse of a Given Matrix

This paragraph concludes the video script by providing an exercise for the viewers to apply the learned method of finding the inverse of a matrix. It presents a new matrix B with elements 2, -3, 5, 6, 0, 1, and 15, and challenges the viewers to find its inverse using the cofactor approach. The paragraph encourages the viewers to share their solutions in the comment section, fostering engagement and interactive learning. It also recaps the process of finding the determinant of matrix A and its inverse, reinforcing the concepts and techniques discussed throughout the video script.

Mindmap
Keywords
πŸ’‘inverse matrix
The inverse matrix is a fundamental concept in linear algebra that represents a matrix which, when multiplied with the original matrix, results in the identity matrix. In the context of the video, the main goal is to find the inverse of a given 3x2 matrix, which is crucial for solving systems of linear equations and understanding the properties of matrix operations.
πŸ’‘cofactor
A cofactor of a matrix is a submatrix that is derived from the original matrix by removing a row and a column, and then multiplying by a specific sign factor. In the video, the cofactor method is used to find the elements of the adjugate matrix, which is a key step in calculating the inverse of a matrix. The signs of the cofactors alternate and are determined by the position of the element in the matrix.
πŸ’‘matrix dimensions
Matrix dimensions refer to the size of a matrix, which is defined by the number of rows and columns it contains. Understanding the dimensions of a matrix is essential for matrix operations, including finding the inverse, as it dictates the possible size and shape of the resulting matrices.
πŸ’‘determinant
The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix and is used to find the inverse of a matrix. It is important because it helps determine whether a matrix has an inverse; if the determinant is zero, the matrix does not have an inverse.
πŸ’‘adjugate matrix
The adjugate matrix, also known as the adjoint matrix, is the transpose of the cofactor matrix of a square matrix. It plays a significant role in finding the inverse of a matrix because the inverse is obtained by dividing the adjugate matrix by the determinant of the original matrix.
πŸ’‘transpose
Transposing a matrix involves interchanging its rows and columns to create a new matrix. This operation is crucial when finding the adjugate matrix, which is the transpose of the cofactor matrix.
πŸ’‘linear equations
Linear equations are mathematical equations in which the variables have a degree of one. They are fundamental in many areas of mathematics and science, and matrix operations, including finding the inverse of a matrix, are often used to solve systems of linear equations.
πŸ’‘sign pattern
The sign pattern refers to the alternating signs in the cofactor matrix, which is a result of the process of removing rows and columns to form the submatrices for calculating the cofactors. The sign pattern is crucial for constructing the correct cofactor matrix and, ultimately, the adjugate matrix.
πŸ’‘element elimination
Element elimination is the process of removing or nullifying certain elements in a matrix to simplify it for further calculations, such as finding the determinant or the cofactor matrix. This technique is used in the video to create smaller matrices from which the cofactors are calculated.
πŸ’‘matrix multiplication
Matrix multiplication is an operation that takes a pair of matrices and produces a third matrix as the result. It is an essential operation in linear algebra and is used in the process of finding the inverse of a matrix, as the inverse is obtained by multiplying the adjugate matrix by the reciprocal of the determinant.
Highlights

Introduction to finding the inverse of a 3x2 matrix using the cofactor approach.

Explanation of the matrix setup and identification of matrix A with its elements.

Detailed process on how to solve for the cofactor of matrix A, emphasizing the alternating sign rule for cofactors.

Computation of the cofactor for the first element in the matrix, showing the elimination process and determinant calculation.

Explanation of the process to compute the determinant of a reduced 2x2 matrix from matrix A.

Calculation and adjustment of signs for the second element's cofactor, resulting in a positive six.

Determination of the cofactor for the third position in matrix A with a resultant positive three.

Methodology for computing the cofactors of the second row of matrix A, highlighting determinant findings.

In-depth analysis of the determinant calculation for various matrix sub-sections, showcasing negative and positive adjustments.

Introduction of the adjoint matrix (adint of mat A), explaining the transformation from cofactors to the adjoint by transposing rows to columns.

Overview of calculating the inverse of matrix A by using the determinant and the adjoint matrix.

Explanation of determinant calculation for matrix A using the cofactor expansion method.

Comprehensive guide on transforming the determinant results into the final inverse matrix expression.

Summary of the mathematical processes and logic used throughout the calculation of the matrix inverse.

Closing remarks and an invitation to viewers to attempt solving for the inverse of a different matrix using the explained cofactor approach.

Transcripts
Rate This

5.0 / 5 (0 votes)

Thanks for rating: