Determinants - Matrix inverse using determinants

• In algebra, the determinant is a function that associates a scalar, determinant, to every square matrix.
• It plays a vital role in solving various problems in linear algebra. "

Determinant of a Matrix

• The determinant of a matrix is denoted by |A| or det(A).

• For a 2x2 matrix A = [a b; c d], its determinant is given by: |A| = ad - bc.

• For a 3x3 matrix A = [a b c; d e f; g h i], its determinant is given by: |A| = a(ei - fh) - b(di - fg) + c(dh - eg).

  Example: For the matrix A = [2 3; 4 1], the determinant is |A| = (21) - (34) = -10. "

Properties of Determinants

1. If A is a square matrix, then det(A^T) = det(A).

2. If A and B are nxn matrices, then det(AB) = det(A) * det(B).

3. If A is an invertible matrix, then det(A^(-1)) = 1 / det(A).

4. If A is a triangular matrix, then the determinant is the product of its diagonal elements. Example: For the matrix A = [3 0 0; 2 4 0; 1 2 5], the determinant is |A| = (345) = 60.

5. If A is a square matrix with a row or column of zeros, then det(A) = 0. "

Finding the Inverse of a Matrix using Determinants

• The inverse of a matrix A, denoted as A^(-1), is a matrix having the property that A * A^(-1) = I, where I is the identity matrix.
• The inverse of a matrix can be computed using determinants.
• For a square matrix A, the inverse is given by: A^(-1) = (1 / det(A)) * Adj(A), where Adj(A) represents the adjugate of matrix A. "

Steps to Find the Inverse of a Matrix

1. Let A be an n x n matrix.

2. Compute the determinant of matrix A: |A|.

3. If |A| = 0, then A^(-1) does not exist.

4. If |A| ≠ 0, compute the adjugate of A: Adj(A).

5. Calculate the inverse of matrix A using the formula: A^(-1) = (1 / |A|) * Adj(A). Example: For the matrix A = [3 1; 4 2], the determinant |A| = (32) - (14) = 2. The adjugate of A, Adj(A) = [2 -4; -1 3]. Therefore, the inverse of A, A^(-1) = (1 / 2) * [2 -4; -1 3] = [1 -2; -0.5 1.5]. "

Properties of Matrix Inverse

• If A is an invertible matrix, then (A^(-1))^(-1) = A.
• If A and B are invertible matrices, then (AB)^(-1) = B^(-1) * A^(-1).
• If A is an invertible matrix, then (A^(-1))^T = (A^T)^(-1).
• If A is an invertible matrix, then det(A^(-1)) = 1 / det(A). Example: If A = [2 1; 3 4] and B = [1 0; -2 3], then A^(-1) = [4 -1; -3 2] and B^(-1) = [1 0; 2/3 1/3]. Therefore, (AB)^(-1) = [1 0; 0 1] = I. "

Solving Linear Equations using Inverse Matrix

• The inverse of a matrix can be used to solve a system of linear equations.
• Consider the equation AX = B, where A is a given matrix, X is the unknown vector, and B is the constant vector.
• If A is invertible, then the solution is given by X = A^(-1) * B. Example: Solve the equation AX = B, where A = [2 1; 3 4] and B = [8 19]. The inverse of A, A^(-1) = [4 -1; -3 2]. Therefore, X = [4 -1; -3 2] * [8 19] = [7 15]. "

Test Yourself

1. Find the determinant of the matrix A = [5 2; -3 1].

2. Calculate the inverse of the matrix A = [2 3; 4 5].

3. Solve the system of equations AX = B, where A = [1 2; 3 4] and B = [7 10]. Remember to revise the properties of determinants and matrix inverse for better understanding.

11. Matrix Inverse and Elementary Row Operations

• Elementary row operations can be used to find the inverse of a matrix.

• Three types of elementary row operations:

  1. Swapping two rows of a matrix
  2. Multiplying a row by a non-zero constant
  3. Adding a multiple of one row to another row

  Example: For the matrix A = [1 2; 3 4], we can find the inverse using elementary row operations.

12. Gauss-Jordan Elimination Method

• The Gauss-Jordan elimination method is an efficient way to find the inverse of a matrix.
• It involves transforming the given matrix into the identity matrix by applying elementary row operations.
• The original matrix is then transformed into its inverse matrix.

13. Steps for Gauss-Jordan Elimination Method

1. Augment the given matrix with the identity matrix.

2. Perform elementary row operations to transform the left part of the augmented matrix into the identity matrix.

3. The right part of the augmented matrix will be the inverse of the original matrix if the left part becomes the identity matrix.

14. Example: Gauss-Jordan Elimination Method

• Consider the matrix A = [3 1; 2 4].
• Augment A with the identity matrix: [3 1 | 1 0; 2 4 | 0 1].
• Perform elementary row operations until the left part becomes the identity matrix.
• The right part of the augmented matrix will be the inverse of A: [3 1 | 1 0; 2 4 | 0 1] -> [1 0 | 4 -1; 0 1 | -1/2 3/2].
• Therefore, the inverse of A is [4 -1; -1/2 3/2].

15. Properties of Inverse Matrix

• If A is an invertible matrix, then the inverse of A is unique.
• If A is an invertible matrix, then AA^(-1) = A^(-1)A = I, where I is the identity matrix.
• If A is an invertible matrix, then (kA)^(-1) = (1/k)A^(-1), where k is a non-zero constant. Example: If A = [2 3; 4 5], then A^(-1) = [5 -3; -4 2]. Therefore, AA^(-1) = [2 3; 4 5] * [5 -3; -4 2] = [1 0; 0 1] = I.

16. Transpose of Inverse Matrix

• The transpose of the inverse of a matrix is equal to the inverse of the transpose of the matrix.
• In other words, if A is an invertible matrix, then (A^T)^(-1) = (A^(-1))^T. Example: If A = [1 2; 3 4], then A^(-1) = [4 -2; -3 1]. The transpose of A, A^T = [1 3; 2 4]. Therefore, (A^T)^(-1) = [1 3; 2 4] ^ (-1) = [4 -2; -3 1].

17. Determinant of Inverse Matrix

• The determinant of the inverse of a matrix is equal to the reciprocal of the determinant of the original matrix.
• In other words, if A is an invertible matrix, then det(A^(-1)) = 1 / det(A). Example: If A = [2 1; 3 4], then det(A) = (24) - (13) = 5. Therefore, det(A^(-1)) = 1 / 5.

18. Non-Invertible Matrices

• Not all matrices have inverses.
• A matrix A is non-invertible (singular) if its determinant, det(A), equals zero.
• In this case, the matrix is said to be singular or degenerate. Example: If A = [1 2; 2 4], then det(A) = (14) - (22) = 0. Therefore, A does not have an inverse.

19. Non-Square Matrices

• Non-square matrices do not have inverses.
• Inverse matrices exist only for square matrices.
• A square matrix is a matrix with an equal number of rows and columns. Example: If A = [1 2; 3 4; 5 6], then A is not a square matrix and does not have an inverse.

20. Summary

• The inverse of a matrix can be found using determinants and elementary row operations.
• Gauss-Jordan elimination method is an efficient way to find the inverse of a matrix.
• The inverse of a matrix has several important properties, such as uniqueness and matrix multiplication rules.
• Not all matrices have inverses, and inverses only exist for square matrices.

Determinants - Matrix inverse using determinants

21. Example: Determinant Calculation

• Let’s calculate the determinant of a 3x3 matrix A = [1 2 3; 4 5 6; 7 8 9].
• The formula for determining the determinant of a 3x3 matrix is: |A| = a(ei - fh) - b(di - fg) + c(dh - eg).
• So, for matrix A, the determinant is: |A| = 1(59 - 68) - 2(49 - 67) + 3(48 - 57).
• Simplifying the equation, we get: |A| = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0.
• Therefore, the determinant of matrix A is 0.

22. Non-Invertible Matrix Determinant

• If a matrix has a determinant of 0, it is called a non-invertible matrix or a singular matrix.
• Non-invertible matrices do not have an inverse because their determinant is zero.
• In other words, the system of equations represented by a non-invertible matrix does not have a unique solution.
• Example: Consider a matrix A = [2 4; 1 2], where |A| = (22) - (41) = 0. Therefore, matrix A is non-invertible.

23. Singular vs. Nonsingular Matrices

• A singular matrix, also known as a non-invertible or degenerate matrix, has a determinant of zero.
• A nonsingular matrix, also known as an invertible or non-degenerate matrix, has a nonzero determinant.
• Nonsingular matrices have inverses, while singular matrices do not.
• Nonsingular matrices represent systems of equations that have a unique solution, while singular matrices represent systems with no unique solution.

24. Applications of Matrix Inverse

• Matrix inverses have various applications in different fields of mathematics and engineering.
• They are used for solving systems of linear equations.
• Inverse matrices are used in cryptography for encryption and decryption algorithms.
• They play a crucial role in solving problems in linear transformations and eigenvalues.
• Matrix inverses are also used in computer graphics and image processing.

25. Pseudoinverse of a Matrix

• In some cases, a general rectangular matrix may not have an inverse.
• But we can still define a pseudoinverse matrix for such cases.
• The pseudoinverse of a matrix A is denoted as A^+ and it satisfies the property: A * A^+ * A = A.
• The pseudoinverse can be computed using techniques such as Singular Value Decomposition (SVD).
• Pseudoinverses are used in statistics, data analysis, control systems, and other areas.

26. Example: Pseudoinverse Calculation

• Suppose we have a rectangular matrix A = [1 2 3; 4 5 6].
• The pseudoinverse of A, denoted as A^+, is computed using SVD.
• The SVD of A is given by A = U * Σ * V^T, where U and V are orthogonal matrices, and Σ is a diagonal matrix.
• The pseudoinverse is calculated as A^+ = V * Σ^+ * U^T, where Σ^+ is the reciprocal of the nonzero diagonal elements of Σ.
• Applying the formula, we get A^+ = [0.1 -0.2; 0.2 -0.4; 0.3 -0.6].

27. Properties of Pseudoinverse

• The pseudoinverse of a matrix A has the following properties:
  1. A * A^+ * A = A
  2. A^+ * A * A^+ = A^+
  3. (A * A^+)^T = A * A^+
  4. (A^+ * A)^T = A^+ * A
• These properties make the pseudoinverse useful in least square solutions, where an exact solution is not possible.

28. Moore-Penrose Pseudoinverse

• The Moore-Penrose pseudoinverse is a specific type of pseudoinverse.
• It is the most commonly used pseudoinverse and is defined for any matrix, whether it is rectangular or square.
• The Moore-Penrose pseudoinverse satisfies all the properties of a pseudoinverse.
• It is named after E. H. Moore and A. N. Kolmogorov, who independently introduced the concept.

29. Calculating Moore-Penrose Pseudoinverse

• The Moore-Penrose pseudoinverse of a matrix A can be calculated as follows:
  1. Compute the SVD of A: A = U * Σ * V^T.
  2. Calculate the pseudoinverse as A^+ = V * Σ^+ * U^T, where Σ^+ is the reciprocal of the nonzero singular values of Σ.
• The pseudoinverse matrix A^+ allows for finding a “best fit” solution when solving an over-determined system of linear equations.

30. Recap

• Determinants play a vital role in finding the inverse of a matrix.
• Invertible matrices have unique inverses, while non-invertible matrices have a determinant of zero.
• Matrix inverses have applications in solving systems of linear equations and cryptography.
• Pseudoinverses are used for rectangular matrices and satisfy some properties of inverse matrices.
• The Moore-Penrose pseudoinverse is a commonly used pseudoinverse.