Determinant of Inverse Matrix

10 Key excerpts on "Determinant of Inverse Matrix"

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  eBook - ePub

  Introduction to Linear Algebra

  • Rita Fioresi, Marta Morigi(Authors)
  • 2021(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  CHAPTER 7

  Determinant and Inverse

  In this chapter, we introduce two basic concepts: the determinant and the inverse of a square matrix. The importance of these two concepts will be summarized by Theorem 7.6.1 , which contains essentially all we learnt about the linear maps from ℝ

  n

  to ℝ

  n

  .

  7.1 DEFINITION OF DETERMINANT

  The concepts of linear algebra introduced so far are not sufficient to give a direct definition of the determinant. Hence, we shall introduce the determinant with an indirect definition and then we will compute it in some special cases, which will prove to be the most significant for us, and finally we will arrive at an algorithmic method to compute it in general.

  Let A be a square matrix, i.e. an

  n × n

  matrix. We want to associate to it a real number, called determinant of A, which is calculated starting from the elements of the matrix A. The definition we give is apparently not a constructive one, however, we will see that, starting from simple rules, we can calculate the determinant of a matrix.

  Before we begin, it is necessary to introduce the definition of the identity matrix.

  Definition 7.1.1 The identity matrix or identity matrix of order n, is the

  n × n

  matrix having all the elements of the main diagonal equal to the number 1, while the remaining elements are equal to 0. Usually it is indicated with I

  n

  , or with I, if there are no ambiguities.

  For example, the identity matrix of order 3 is:

  I =

  1 0 0

  0 1 0

  0 0 1

  .

  Definition 7.1.2 The determinant of a square matrix A of order n is a real number, denoted by

  det ⁡ ( A )

  , with the following properties:

  1. If the j-th row of A is the sum of two elements u and v of ℝ

     n

     , then the determinant of A is the sum of the determinants of the two matrices obtained by replacing the j-th row of A with u and v, respectively.
  2. If the j-th row of A is the product

     λ u

     , where u is an element of ℝ

     n

     and λ is a scalar, then the determinant of A is the product of λ and the determinant of the matrix obtained by replacing the j-th row of A with u

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  Linear Algebra and Matrix Analysis for Statistics

  • Sudipto Banerjee, Anindya Roy(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  CHAPTER 10 Determinants 10.1 Introduction Sometimes a single number associated with a square matrix can provide insight about the properties of the matrix. One such number that we have already encountered is the trace of a matrix. The “determinant” is another example of a single number that conveys an amazing amount of information about the matrix. For example, we will soon see that a square matrix will have no inverse if and only if the determinant is zero. Solutions to linear systems can also be expressed in terms of determinants. In multivariable calculus, probability and statistics, determinants arise as “Jacobians” when transforming variables by differentiable functions. A formal definition of determinants uses permutations . We have already encountered permutations in the definition of permutation matrices. More formally, we can define the permutation as follows. Definition 10.1 A permutation is a one-one map from a finite non-empty set S onto itself. Let S = { s 1 , s 2 . . . , s n } . A permutation π is often represented as a 2 × n array π = s 1 s 2 · · · s n π ( s 1 ) π ( s 2 ) · · · π ( s n ) . (10.1) For notational convenience, we will often write π ( i ) as π i . The finite set S of n elements is often taken as (1 , 2 , . . . , n ) , in which case we write π as π = ( π 1 , π 2 , ..., π n ) . Simply put, a permutation is any rearrangement of (1 , 2 , . . . , n ) . For example, the complete set of permutations for { 1 , 2 , 3 } is given by { (1 , 2 , 3) , (1 , 3 , 2) , (2 , 1 , 3) , (2 , 3 , 1) , (3 , 1 , 2) , (3 , 2 , 1) } . In general, the set (1 , 2 , . . . , n ) has n ! = n · ( n -1) · · · 2 · 1 different permutations. Example 10.1 Consider the following two permutations on S = (1 , 2 , 3 , 4 , 5) : π = (1 , 3 , 4 , 2 , 5) and θ = (1 , 2 , 5 , 3 , 4) . Suppose we apply θ followed by π to (1 , 2 , . . . , 5) . Then the resultant permutation is is given by πθ = ( π θ 1 , π θ 2 , . . . , π θ 5 ) = ( π 1 , π 2 , π 5 , π 3 , π 4 ) = (1 , 3 , 5 , 4 , 2) .

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  Algebra & Trig

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  You will study determinants in the next section. EXAMPLE 4 Finding the Inverse of a 2 × 2 Matrix See LarsonPrecalculus.com for an interactive version of this type of example. If possible, find the inverse of each matrix. a. A = [ 3 -2 -1 2 ] b. B = [ 3 -6 -1 2 ] Solution a. The determinant of matrix A is ad - bc = 3(2) - (-1)(-2) = 4. This quantity is not zero, so the matrix is invertible. The inverse is formed by interchanging the entries on the main diagonal, changing the signs of the other two entries, and multiplying by the scalar 1 4 . A -1 = 1 ad - bc [ d -c -b a ] Formula for the inverse of a 2 × 2 matrix = 1 4 [ 2 2 1 3 ] Substitute for a, b, c, d, and the determinant. = [ 1 2 1 2 1 4 3 4 ] Multiply by the scalar 1 4 . b. The determinant of matrix B is ad - bc = 3(2) - (-1)(-6) = 0. Because ad - bc = 0, B is not invertible. Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com If possible, find the inverse of A = [ 5 3 -1 4 ] . Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 10.3 The Inverse of a Square Matrix 733 GO DIGITAL Systems of Linear Equations You know that a system of linear equations can have exactly one solution, infinitely many solutions, or no solution. If the coefficient matrix A of a square system (a system that has the same number of equations as variables) is invertible, then the system has a unique solution, which can be found using an inverse matrix as follows.

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  Linear Algebra: Gateway to Mathematics

  Second Edition

  • Robert Messer(Author)
  • 2021(Publication Date)
  • American Mathematical Society

    (Publisher)

  In this project you are invited to consider alternative definitions of the determinant of a square matrix. One popular definition of determinants follows from some preliminary work on permutation functions. You will discover that the main results about determinants are closely related to some simple properties of permutations. Since permutations are useful in other contexts, you will be doubly rewarded when you run across them as old friends in subsequent work in mathematics. 1. A permutation of a set ? is a function 𝜎 ∶ ? → ? that is one-to-one and onto. In this project we need only consider permutations defined on finite sets of the form {1, 2, ... , ?} where ? is a positive integer. The ordered set {𝜎(1), 𝜎(2), ... , 𝜎(?)} is a convenient way to specify a permutation 𝜎 defined on {1, 2, ... , ?} . Use this notation to write down all permutations of {1, 2, 3} . How many permutations are there on a set of ? elements? Given a permutation 𝜎 defined on a set {1, 2, ... , ?} , define an inversion to be a pair of integers 𝑖 and 𝑗 in {1, 2, ... , ?} with 𝑖 < 𝑗 and 𝜎(𝑖) > 𝜎(𝑗) . Find a systematic method of counting the number of inversions of a permutation 𝜎 . Determine the number of inversions of each of the permutations of {1, 2, 3} . What are the minimum and maximum numbers of inversions possible for a permutation of {1, 2, ... , ?} ? Define a permutation to be even if and only if the number of inversions is even. Similarly define an odd permutation. Define a transposition to be a permutation that interchanges two elements and leaves the other elements fixed. Suppose 𝜎 is any permutation of {1, 2, ... , ?} and 𝜏 is a transposition defined on {1, 2, ... , ?} . Show that the number of inversions of 𝜎 and the number of inversions of 𝜏∘𝜎 differ by 1 . You might want to consider first the case where 326 Chapter 7. Determinants 𝜏 is a transposition of adjacent elements. Show that any permutation on {1, 2, ... , ?} can be written as a composition of transpositions.

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  Precalculus with Limits

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 8.4 The Determinant of a Square Matrix 577 GO DIGITAL 8.4 The Determinant of a Square Matrix Find the determinants of 2 × 2 matrices. Find minors and cofactors of square matrices. Find the determinants of square matrices. The Determinant of a 2 × 2 Matrix Every square matrix can be associated with a real number called its determinant. Determinants have many uses, and several will be discussed in this section and the next section. Historically, the use of determinants arose from special number patterns that occur when systems of linear equations are solved. For example, the system { a 1 x a 2 x + + b 1 y b 2 y = = c 1 c 2 has a solution x = c 1 b 2 - c 2 b 1 a 1 b 2 - a 2 b 1 and y = a 1 c 2 - a 2 c 1 a 1 b 2 - a 2 b 1 provided that a 1 b 2 - a 2 b 1 ≠ 0. Note that the denominators of the two fractions are the same. This denominator is called the determinant of the coefficient matrix of the system. Coefficient Matrix Determinant A = [ a 1 a 2 b 1 b 2 ] det(A) = a 1 b 2 - a 2 b 1 The determinant of matrix A can also be denoted by vertical bars on both sides of the matrix, as shown in the definition below. In this text, det(A) and ∣ A ∣ are used interchangeably to represent the determinant of A. Although vertical bars are also used to denote the absolute value of a real number, the context will show which use is intended. A convenient method for remembering the formula for the determinant of a 2 × 2 matrix is shown below. det(A) = ∣ a 1 a 2 b 1 b 2∣ = a 1 b 2 - a 2 b 1 Note that the determinant is the difference of the products of the two diagonals of the matrix. Determinants are often used in other branches of mathematics.

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  Algebraic Number Theory for Beginners

  Following a Path From Euclid to Noether

  • John Stillwell(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  7 Determinant Theory Preview The determinant function is a concept of linear algebra frequently skimmed over or minimized in modern treatments of the subject. At the same time, books on algebraic number theory tend to assume sophisticated properties of the determinant – such as its relationship to trace, norm, and characteristic polynomial – to be already known from a basic linear algebra course. Under these circumstances it seems useful to develop the determinant concept from scratch and then transition to its applications in algebraic number fields. This is what we aim to do in this chapter, hopefully making the book more self-contained without greatly increasing its size. We begin with an elementary treatment of determinants, due to Artin (1942). His approach leads rapidly to methods for evaluating determinants, applications to linear equations, and to the all-important multiplicative property. We can then prove the invariance of the determinant under change of basis and deduce the basis-independence of trace, norm, and characteristic polynomial. This leads in turn to relations between the trace and norm of an algebraic number and the roots of its minimal polynomial. With these foundations we can then introduce the discriminant, which tests whether an n-tuple of members of a field F of degree n over Q is a basis for F . This paves the way for the study of integral bases in the next chapter. These generalize the concept of basis from vector spaces to certain kinds of modules, such as the algebraic number rings Z E . 149 150 7 Determinant Theory Figure 7.1 Emil Artin (1898–1962) (used with permission of Tom Artin and his siblings). 7.1 Axioms for the Determinant There are many ways to define the determinant function and derive its basic properties, none of them completely straightforward. One that is elementary, yet algebraic in spirit, is given in Artin (1942), pages 12–20.

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  College Algebra

  • James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 538 CHAPTER 6 ■ Matrices and Determinants Determinant of an n  n Matrix (p. 526) To find the determinant of the n  n matrix A  D a 11 a 12 c a 1 n a 21 a 22 c a 2 n ( ( f ( a n 1 a n 2 c a nn T we choose a row or column to expand , and then we calculate the number that is obtained by multiplying each element of that row or column by its cofactor and then adding the resulting products. For example, if we choose to expand about the first row, we get det 1 A 2  0 A 0  a 11 A 11  a 12 A 12  . . .  a 1 n A 1 n Invertibility Criterion (p. 527) A square matrix has an inverse if and only if its determinant is not 0. Row and Column Transformations (p. 528) If we add a nonzero multiple of one row to another row in a square matrix or a nonzero multiple of one column to another column, then the determinant of the matrix is unchanged. Cramer’s Rule (pp. 529–531) If a system of n linear equations in the n variables x 1 , x 2 , c , x n is equivalent to the matrix equation DX  B and if 0 D 0 ? 0 , then the solutions of the system are x 1  0 D x 1 0 0 D 0 x 2  0 D x 2 0 0 D 0 . . . x n  0 D x n 0 0 D 0 where D x i is the matrix that is obtained from D by replacing its i th column by the constant matrix B . Area of a Triangle Using Determinants (p. 532) If a triangle in the coordinate plane has vertices 1 a 1 , b 1 2 , 1 a 2 , b 2 2 , and 1 a 3 , b 3 2 , then the area of the triangle is given by !   1 2 3 a 1 b 1 1 a 2 b 2 1 a 3 b 3 1 3 where the sign is chosen to make the area positive. 1. What does it mean to say that A is a matrix with dimension m  n ? 2. What is the row-echelon form of a matrix? What is a leading entry? 3.

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  An Introduction to Differential Equations

  Deterministic Modeling, Methods and Analysis(Volume 1)

  • Anil G Ladde, G S Ladde;;;(Authors)
  • 2012(Publication Date)
  • WSPC

    (Publisher)

  Chapter 1 Elements of Matrices, Determinants, and Calculus 1.1 Introduction This chapter serves as a review. It begins by highlighting a few ideas about the process of solving mathematics/education/research problems, and then covers rel-evant mathematical concepts and statements in linear algebra and the calculus of matrix functions. In particular, the algebra of matrices, the properties of deter-minants, some concepts about vector spaces, sets of independent and dependent vectors, methods for solving systems of linear-algebraic equations, and Wronskians of functions are discussed in Sections 1.3 and 1.4. Moreover, differential calculus of determinant functions, particularly a generalized mean-value theorem and Taylor’s formula for determinant functions, are developed in Section 1.5. These results play a very important role in finding solutions to linear differential equations, especially stochastic differential equations. Note that we are not attempting to teach linear algebra and multivariate calculus. This chapter not only serves as a reference guide for the reader, but it also is teaching guide for the instructor. 1.2 Problem-Solving Process One of the most important goals of education is to gain knowledge about the entire universe, and to apply it for the benefit of living things on the Earth. Its purpose is also to gain knowledge of the problem-solving process to eradicate or understand our ignorance. The presented knowledge about the problem-solving process can easily be applied in any area of science: biological, chemical, engineering, mathematical, medical, physical, political, social, etc. Currently, it is well recognized that the knowledge and practices employed in the mathematical-problem-solving processes provide a suitable background and the tools for solving problems arising in any discipline. In the 21st century, the question that is important to all of us is how to gain knowledge of the complex system (the 1

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  Linear Algebra: Concepts and Methods

  • Martin Anthony, Michele Harvey(Authors)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Matrix inversion and determinants In this chapter, all matrices will be square n × n matrices, unless explic- itly stated otherwise. Only a square matrix can have an inverse, and the determinant is only defined for a square matrix. We want to answer the following two questions: When is a matrix A invertible? How can we find the inverse matrix? 3.1 Matrix inverse using row operations 3.1.1 Elementary matrices Recall the three elementary row operations: RO1 multiply a row by a non-zero constant. RO2 interchange two rows. RO3 add a multiple of one row to another. These operations change a matrix into a new matrix. We want to exam- ine this process more closely. Let A be an n × n matrix and let A i denote the i th row of A. Then we can write A as a column of n rows, A = ⎛ ⎜ ⎜ ⎜ ⎝ a 11 a 12 · · · a 1n a 21 a 22 · · · a 2n . . . . . . . . . . . . a n1 a n2 · · · a nn ⎞ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎝ A 1 A 2 . . . A n ⎞ ⎟ ⎟ ⎟ ⎠ . 3.1 Matrix inverse using row operations 91 We use this row notation to indicate row operations. For example, what row operations are indicated below? ⎛ ⎜ ⎜ ⎜ ⎝ A 1 3 A 2 . . . A n ⎞ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ A 2 A 1 . . . A n ⎞ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ A 1 A 2 + 4 A 1 . . . A n ⎞ ⎟ ⎟ ⎟ ⎠ . The first is multiply row 2 by 3, the second is interchange row 1 and row 2, and the third is add 4 times row 1 to row 2. Each of these represents new matrices after the row operation has been executed. Now look at a product of two n × n matrices A and B . The (1, 1) entry in the product is the inner product of row 1 of A and column 1 of B . The (1, 2) entry is the inner product of row 1 of A and column 2 of B , and so on. In fact, row 1 of the product matrix AB is obtained by taking the product of the row A 1 with the matrix B ; that is, A 1 B . This is true of each row of the product; that is, each row i of the product AB is obtained by calculating A i B . So we can express the product AB as ⎛ ⎜ ⎜ ⎜ ⎝ a 11 a 12 · · · a 1n a 21 a 22 · · · a 2n .

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  Linear Algebra and Geometry

  • Francesco Bottacin(Author)
  • 2023(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  Definition 3.2.22. Given a matrix A = (a ij ) ∈ M n (K), for any i, j ∈ {1, . . . , n} we denote by A ij the matrix of order n − 1 obtained from A by deleting its i-th row and j -th column. The determinant of the matrix A ij is called the minor of the matrix A corresponding to the indices i and j . The expression a ∗ ij = (−1) i+j |A ij | is called the cofactor of the element a ij of A. The transpose of the matrix formed by the cofactors of the elements of A is called the adjoint matrix of A, and will be denoted by adj(A) or A ∗ : A ∗ = adj(A) = t ( a ∗ ij ) ∈ M n (K). We can now prove the following result, which provides a very useful method for computing the determinant of a matrix. Proposition 3.2.23 (Laplace formula). Let A ∈ M n (K). For every row index i we have: det A = n  h=1 (−1) i+h a ih |A ih |. (3.2.1) Similarly for every column index j we have: det A = n  k=1 (−1) k+j a kj |A kj |. (3.2.2) Proof. Since the determinant of a matrix coincides with that of its transpose, exchanging the roles of the rows and columns of the matrix A the formula (3.2.1) reduces to (3.2.2). Therefore it suffices to prove one of the two for- mulas, for example (3.2.1). 3.2. The determinant of a square matrix 105 We now want to prove that, in fact, it is enough to prove the formula (3.2.1) for i = 1. Let us assume that Laplace formula holds for the first row (i.e., for i = 1). Let us choose a row index i > 1 and write the matrix A as A = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ A (1) . . . A (i−1) A (i) A (i+1) . . . A (n) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ With i − 1 exchanges of contiguous rows it is possible to bring the i-th row of A in the place of the first row, obtaining the matrix A  = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ A (i) A (1) . . . A (i−1) A (i+1) . . . A (n) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Since the determinant changes its sign whenever we swap two rows, we have det A = (−1) i−1 det A  .

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