Inverse of a 3x3 Matrix

The Organic Chemistry Tutor
18 Feb 201815:21
EducationalLearning
32 Likes 10 Comments

TLDRThis video tutorial provides a step-by-step guide on how to calculate the inverse of a 3x3 matrix. It begins by explaining the need to form an augmented matrix with the given matrix and its multiplicative identity. The process involves performing elementary row operations to transform the left side of the augmented matrix into the identity matrix, while the right side becomes the inverse. The video meticulously demonstrates each row operation, including subtraction and multiplication, to achieve the desired outcome. Finally, it confirms the correctness of the calculated inverse by multiplying it with the original matrix, resulting in the identity matrix, thereby validating the method and providing the viewer with a comprehensive understanding of matrix inversion.

Takeaways
  • 📌 To find the inverse of a 3x3 matrix, start by forming an augmented matrix with the given matrix and the multiplicative identity matrix of the same size.
  • 🔄 Perform elementary row operations on the augmented matrix to transform the left side (original matrix) into the identity matrix, while applying the same operations to the right side (which will become the inverse matrix).
  • ➖ The first step in the row operations is to make the first element of the original matrix's first row into a 1 by subtracting the third row from the first row.
  • 🔄 Following the first step, make the second element of the first row into a 0 by applying the same operation that was used on the first row (subtraction) to the second row.
  • ➗ To make the second element of the second row into a 0, multiply the first row by 2 and add it to the second row, changing the second row accordingly.
  • 🔄 To get the third row's second element to zero, multiply the second row by 3 and subtract it from the third row.
  • 🔄 Continue with row operations until the original matrix's side of the augmented matrix becomes the identity matrix, and the right side will be the inverse matrix.
  • 🔢 After obtaining the inverse matrix, confirm its validity by multiplying the original matrix with the inverse matrix and checking if the result is the identity matrix.
  • 📈 Each step in the row operations should be carefully executed, ensuring that the same changes are applied to both sides of the augmented matrix.
  • 🎓 Understanding the process of finding the inverse of a 3x3 matrix is crucial for various mathematical and computational applications.
  • 👍 The method demonstrated in the video is systematic and can be applied to any 3x3 matrix to find its inverse, provided the matrix is invertible.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is how to determine the inverse of a 3x3 matrix.

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

    -The first step is to rewrite the matrix in the form of an augmented matrix with the multiplicative identity of a 3x3 matrix.

  • What does the multiplicative identity matrix of a 3x3 matrix look like?

    -The multiplicative identity matrix of a 3x3 matrix is a matrix with ones on the diagonal (1, 0, 0; 0, 1, 0; 0, 0, 1) and zeros elsewhere.

  • What are elementary row operations?

    -Elementary row operations are basic row manipulations such as swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another.

  • Why are elementary row operations used in finding the inverse of a matrix?

    -Elementary row operations are used to transform the left side of the augmented matrix into the identity matrix, while the right side changes accordingly, ultimately giving us the inverse matrix.

  • How do you ensure that the same operations applied to the left side of the augmented matrix are also applied to the right side?

    -You perform the same row operations on the right side as you do on the left side to maintain the augmented matrix structure and ensure the final result on the right side is the inverse matrix.

  • What is the purpose of transforming the left side of the augmented matrix into the identity matrix?

    -Transforming the left side into the identity matrix is a method to find the inverse of the original matrix. When the left side becomes the identity matrix, the right side becomes the inverse of the original matrix.

  • How do you confirm that a matrix is indeed the inverse of another matrix?

    -To confirm that a matrix is the inverse, you multiply the original matrix by the candidate inverse matrix and show that the result is the identity matrix.

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

    -The final step is to verify the result by multiplying the original matrix with the calculated inverse matrix and confirming that the product is the identity matrix.

  • What happens if the product of the matrix and its inverse is not the identity matrix?

    -If the product is not the identity matrix, then the candidate matrix is not the inverse of the original matrix, and the process should be reviewed for errors or alternative methods should be considered.

  • Why is finding the inverse of a matrix important?

    -Finding the inverse of a matrix is important in various mathematical applications, including solving systems of linear equations, calculating the determinant, and understanding matrix properties.

Outlines
00:00
📚 Introduction to Finding the Inverse of a 3x3 Matrix

This paragraph introduces the concept of finding the inverse of a 3x3 matrix. It explains the process of determining the inverse of a given matrix 'a' by first rewriting it in the form of an augmented matrix, combining the matrix with the multiplicative identity of the same size. The paragraph outlines the need to perform elementary row operations to transform the left side of the augmented matrix, ensuring that the same operations are applied to the right side. The goal is to make the left side resemble the identity matrix, with the resulting right side being the inverse of the original matrix.

05:08
🔢 Step-by-Step Row Operations for Matrix Inversion

This paragraph delves into the specifics of the row operations required to invert the 3x3 matrix. It details the process of turning specific elements into zeros and the corresponding operations needed to achieve this. The paragraph walks through the calculations, such as subtracting and adding multiples of rows, to manipulate the matrix into the desired form. It emphasizes the importance of applying the same transformations to both sides of the augmented matrix to eventually obtain the inverse on the right side.

10:11
📈 Verification of the Inverse Matrix

The final paragraph focuses on verifying the accuracy of the calculated inverse matrix. It explains the need to multiply the original matrix 'a' by its calculated inverse and demonstrate that the result is the multiplicative identity matrix. The paragraph provides a step-by-step verification process, illustrating how to multiply each row of the original matrix by each column of the inverse and summing the products to ensure they result in the identity matrix's values. This verification confirms the correctness of the inverse matrix and wraps up the explanation on how to find the inverse of a 3x3 matrix.

Mindmap
Keywords
💡Matrix
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of the video, Matrix A is a 3x3 matrix with specific elements that the user wants to find the inverse of. The process of finding the inverse is central to the video's educational content, as it demonstrates the steps needed to manipulate the matrix to achieve a desired result, which is the inverse matrix in this case.
💡Augmented Matrix
An augmented matrix is a combination of a matrix and its associated matrix, typically used in solving systems of linear equations or finding the inverse of a matrix. In the video, the augmented matrix is formed by placing Matrix A alongside the multiplicative identity of a 3x3 matrix, which serves as a starting point for the row operations needed to find the inverse.
💡Elementary Row Operations
Elementary row operations are basic operations performed on the rows of a matrix that do not change the solutions of the system of equations represented by the matrix. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another. In the video, these operations are essential for transforming the left side of the augmented matrix into the identity matrix, which in turn reveals the inverse on the right side.
💡Inverse of a Matrix
The inverse of a matrix is another matrix that, when multiplied with the original matrix, results in the identity matrix. The inverse is only existent for square matrices and is used in various mathematical applications, including solving systems of equations and finding the determinant. The video's main objective is to teach the viewer how to calculate the inverse of a 3x3 matrix, which is a fundamental concept in linear algebra.
💡Multiplicative Identity
The multiplicative identity, often denoted by 1 or 'one', is a property in mathematics where a number multiplied by any other number does not change the other number. In the context of matrices, the multiplicative identity is a matrix that, when multiplied by another matrix, leaves the other matrix unchanged. In the video, the identity matrix is used as a reference to transform the given matrix into its inverse through a series of row operations.
💡Row Operations
Row operations are mathematical manipulations performed on the rows of a matrix or an augmented matrix. These operations are fundamental in linear algebra for solving systems of equations, finding the determinant, and calculating the inverse of a matrix. In the video, the detailed process of finding the inverse of a matrix relies heavily on correctly applying row operations to transform the matrix into the identity matrix.
💡Identity Matrix
An identity matrix is a special square matrix with ones on the diagonal and zeros elsewhere. When multiplied by any matrix of the same size, the result is the original matrix. The identity matrix plays a crucial role in matrix operations as it is the equivalent of the number 1 in scalar multiplication. In the video, the identity matrix serves as the target form that the left side of the augmented matrix is transformed into, with the right side becoming the inverse of the original matrix.
💡Linear Equations
Linear equations are mathematical equations in which the highest power of the variable is one. They represent straight lines in a two-dimensional space and can be extended to higher dimensions. In the context of matrices, systems of linear equations can be represented as matrices, and operations on these matrices can be used to solve the equations. The video's content on finding the inverse of a matrix is closely related to solving systems of linear equations, as the inverse matrix can be used to find the solutions.
💡Determinant
The determinant of a square matrix is a scalar value that can be used to find the area or volume of a geometric shape defined by the matrix's rows or columns. It is also an essential concept in determining whether a matrix has an inverse. If the determinant of a matrix is zero, the matrix does not have an inverse. While the video focuses on finding the inverse of a matrix, understanding the concept of determinant is crucial for ensuring the matrix can indeed have an inverse.
💡Linear Algebra
Linear algebra is a branch of mathematics that deals with linear equations, linear transformations, and vector spaces. It is the study of linear systems and their representations using matrices. The video's content on finding the inverse of a 3x3 matrix is a fundamental topic in linear algebra, showcasing the practical application of matrix manipulation techniques to solve problems in various mathematical and real-world scenarios.
💡Scalar
A scalar is a single number that represents a quantity with magnitude but no direction. In the context of matrices, scalar multiplication involves multiplying every element of a matrix by a single value. This concept is important when performing row operations, as it can help in simplifying the matrix or transforming it into a more manageable form for further calculations, such as finding the inverse.
Highlights

The video explains the process of determining the inverse of a 3x3 matrix, a fundamental concept in linear algebra.

Matrix A is presented with its elements, and the goal is to find its inverse.

An augmented matrix is formed by combining Matrix A with the multiplicative identity of a 3x3 matrix.

Elementary row operations are the key technique used to transform the matrix into its inverse.

The first step involves turning a specific element into a zero using row operations.

The process requires adapting row operations to achieve a desired zero in the matrix.

The video demonstrates how to manipulate row two to turn an element into zero.

A crucial step is shown where row three is modified to achieve a zero in a specific position.

The video guides through the transformation of the matrix, focusing on turning elements into zeros and ones in the correct positions.

The final form of the matrix is revealed, showing the inverse of Matrix A.

The video emphasizes the importance of applying the same row operations to both the left and right sides of the augmented matrix.

Verification of the inverse matrix is done by multiplying it with the original matrix and showing the result is the identity matrix.

The video provides a detailed explanation of each step, ensuring the viewer understands the process of finding the inverse of a matrix.

The practical application of matrix inversion is briefly touched upon, highlighting its relevance in various fields.

The video concludes by summarizing the method and confirming that the found inverse is correct.

Transcripts

Browse More Related Video

PreCalculus - Matrices & Matrix Applications (22 of 33) Using Gaussian Elimin. to Find the Inverse

PreCalculus - Matrices & Matrix Applications (23 of 33) Finding the Inverse of a 3x3 Matrix

Classic video on inverting a 3x3 matrix part 2 | Matrices | Precalculus | Khan Academy

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

Inverse of a 2x2 Matrix

How to find the inverse of a 3 by 3 matrix (3 methods you need to know)

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
Matrix InversionLinear AlgebraRow OperationsAugmented MatrixMath TutorialEducational ContentMathematicsInverse CalculationAlgebra TechniquesMathematics EducationMatrix Theory