Determinants

12 Key excerpts on "Determinants"

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  eBook - ePub

  Linear Algebra

  A First Course with Applications to Differential Equations

  • Tom M. Apostol(Author)
  • 2014(Publication Date)
  • Wiley-Interscience

    (Publisher)

  5

  Determinants

  5.1 Introduction

  In many applications of linear algebra to geometry and analysis the concept of a determinant plays a useful role. This chapter studies the basic properties of Determinants and some of their applications.

  Determinants of order two and three were introduced in Chapter 2 as a convenient notation for expressing certain formulas in compact form. We recall that a determinant of order two was defined by the formula

  (5.1)

  Despite similarity in notation, the determinant (written with vertical bars) is conceptually distinct from the matrix (written with square brackets).

  The determinant is a number assigned to the matrix according to Formula (5.1). To emphasize this connection we also write

  In Chapter 2 Determinants of order three were defined in terms of those of order two by the formula

  (5.2)

  This chapter discusses the general case, the determinant of a square matrix of order n for any n ≥ 1. For a 1 × 1 matrix with entry a 11 define the determinant to be that entry: det [a 11 ] = a 11 .

  Our point of view is to treat the determinant as a function that assigns to each square matrix A a number called the determinant of A and denoted by det A . One way to define this function is by an explicit formula generalizing (5.1) and (5.2). This formula is a sum containing n ! products of entries of A . For large n the formula is unwieldy and is rarely used in practice. It seems preferable to study Determinants from another point of view that emphasizes their essential properties. These properties, which are important in the applications, will be taken as axioms for a determinant function.

  Initially, our program will consist of three parts: (1) To motivate the choice of axioms. (2) To deduce further properties of Determinants from the axioms. (3) To prove that for each n

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  Mathematics for Economics and Finance

  • Michael Harrison, Patrick Waldron(Authors)
  • 2011(Publication Date)
  • Routledge

    (Publisher)

  Determinants

  DOI: 10.4324/9780203829998-4

  2.1 Introduction

  In Chapter 1 we defined a matrix, saw a few examples of how systems of linear equations and matrices arise in economic and financial applications, and presented the basic operations and rules of matrix algebra, paying attention to the proofs of many of the matrix properties that were stated. We also catalogued a number of special types of matrix that we will encounter later. In this chapter, we introduce the concept of a determinant, which is important in the theory and solution of systems of linear equations. The determinant also provides us with a means of obtaining an explicit formula for evaluating the inverse of a matrix.

  At a basic level, we can describe a determinant as a real-valued function of a square matrix variable, i.e. a function that associates a real number f(A) with a square matrix A. Formally, we may write f : M → ℝ, where M is the set of square matrices and ℝ is the set of real numbers. However, we require a more precise definition that will also allow us to calculate the numerical value of f(A). Before we give this, we need to be clear on a few preliminary ideas.

  2.2 Preliminaries

  2.2.1 Permutation

  The idea of a permutation of some number, r, of integers from the set of integers {1, 2, ..., n} is fundamental. As discussed on p. xxiii, a permutation is an ordered arrangement of r of the first n integers. The number of such different permutations,

  n

  P r

  , is given by the general result

  n

  P r

  =

  n !

  ( n − r ) !

  (2.1) defined on p. xxiii, from which it follows that

  n

  P n

  , the number of ordered arrangements of all n integers, is n!, since (n – n)! = 0! ≡ 1. Of these n! permutations, the typical permutation, i.e. the jth, may be denoted by

  j 1

   

  j 2

    ...  

  j n

  .

  2.2.2 Inversion

  An inversion in a permutation of n integers occurs whenever a larger integer precedes a smaller one in the permutation. For example, if j1 > j2 , we have one inversion; if j1 > j3

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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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  Matrices in Engineering Problems

  • Marvin Tobias(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  23 C H A P T E R 2 Determinants 2.1 INTRODUCTION The definition of a determinant is derived from the solution of linear algebraic equations. Since the single variable case is trivial, we will begin with the (2X2): a 11 x 1 + a 12 x 2 = c 1 a 21 x 1 + a 22 x 2 = c 2 . (2.1) To eliminate x 2 , we multiply the first equation by a 22 , and the second by a 12 , then subtract the second from the first: (a 11 a 22 − a 12 a 21 )x 1 = (a 22 c 1 − a 12 c 2 ) . (2.2) Equivalently, we may eliminate x 1 (by the same methods): (a 11 a 22 − a 12 a 21 )x 2 = (a 11 c 2 − a 21 c 1 ) . (2.3) The coefficients on both sides of (2.2) and (2.3) can be viewed as expansions via “cross-multiplication” of determinant arrays, as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a 11 a 12 a 21 a 22 + - = (a 11 a 22 − a 12 a 21 ) a 11 c 1 a 21 c 2 + - = (a 11 c 2 − c 1 a 21 ) c 1 a 12 c 2 a 22 + - = (c 1 a 22 − a 12 c 2 ) (2.4) The square arrays in (2.4) “expand,” by the cross—multiplication indicated, to the scalars shown on the right sides. And, we define the determinant in terms of its expansion. Expansions are defined only for square arrays. The result of the expansion is a scalar expression, or numeric value. That is, the determinant is a scalar value. Further, from (2.2) and (2.3), the values of the variables are found as the ratio of these expanded Determinants—all of which are known, given in the problem. 24 2. Determinants Three Variables: a 11 x 1 + a 12 x 2 + a 13 x 3 = c 1 a 21 x 1 + a 22 x 2 + a 23 x 3 = c 2 a 31 x 1 + a 32 x 2 + a 33 x 3 = c 3 .

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  eBook - ePub

  Engineering Mathematics with Applications to Fire Engineering

  • Khalid Khan, Tony Lee Graham(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  4 Determinants and Matrices

  4.1 Background

  Matrices with different dimensions are a standard method for solving a wide range of problems in many disciplines. In science and engineering, when considering particular problems, the derivation of the solutions to these problems can come down to solving a system of linear equations. In this chapter, different methods for solving a linear system of equations will be discussed and these methods involve the concepts of Determinants and matrices.

  4.2 Introduction to Determinants

  Determinants arise naturally in the solution of a set of linear equations. Also they will help solve linear equations using the matrix inversion method (see Section 4.3.6 ). Starting with a general set of two linear simultaneous equations as follows:

  a x + b y = e

  4.1

  4.1

  c x + d y = f

  4.2

  4.2

  where a , b , c , d , e , and f are constants. How do you find the values of x and y ? Since the values of a , b , c , and d are not known, all that can be done is try to eliminate either x or y .

  If one decides to get rid of y , then to make the coefficients of y the same, first multiply Equation 4.1 by d and Equation 4.2 by b :

  a d x + b d y = d e

  4.3

  4.3

  b c x + b d y = b f

  4.4

  4.4

  Then subtracting Equation 4.4 from Equation 4.3 :

  a d x − b c x = d e − b f   ∴   x ( a d − b c ) = d e − b f   ∴   x =

  d e − b f

  a d − b c

  A similar approach, getting rid of x , would give the result as

  y =

  a f − c e

  a d − b c

  Now provided that the denominator term ad − bc ≠ 0, then a symbol can be used to define a 2-by-2 determinant using parallel lines (∣ ∣), as shown next:

  |

  a b

  c d

  |

  ≜ a d − b c

  4.5

  4.5

  Note:The symbol ≜ means “defined as.”

  This definition given by Equation 4.5 is just the difference in product of the diagonals.

  Notice also how these answers for x and y

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  eBook - ePub

  Mathematics for Physicists

  • Brian R. Martin, Graham Shaw(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  9 Determinants, Vectors and Matrices

  In Chapter 8, we introduced vectors as objects associated with a direction in everyday three-dimensional space and showed how they can be discussed using equations for their three components in a given reference frame. Here we shall show how to extend the number of components to define vectors in spaces of more than three dimensions. This leads to the introduction of matrices, which are two-dimensional arrays that enable vectors to be transformed into other vectors. The properties of matrices are discussed in detail and their uses illustrated in, for example, solving simultaneous linear equations. In the following chapter we continue the discussion of matrices, with applications to vibrating systems and to geometry. Firstly, however, we study related quantities called Determinants, which will play a crucial role in this development.

  9.1 Determinants

  These occur in many contexts and we have already met examples in the discussion of vectors in Chapter 8. From (8.16b), the vector product of two vectors a and b in Cartesian co-ordinates has an x-component (ay bz − az by ). Any four quantities aij (i, j = 1, 2) combined in this way can be written in the form of a square array, denoted by Δ2 , called a determinant. This is written in the form

  (9.1)

  where the quantities aij (i, j = 1, 2) are called the elements of the determinant. For example,

  The result, in this case − 5, is called the value of the determinant. It is important to note that the vertical bars in (9.1 ) do not mean that a modulus is to be taken, as this example confirms. Although we have used real numbers for the elements in this example, in general they can be algebraic expressions, real or complex, so the value of the determinant may also be real or complex expressions or numbers.

  Determinants of larger dimensionality can also be constructed. Thus the 3 × 3 determinant

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  eBook - ePub

  Linear Algebra

  • Reg Allenby(Author)
  • 1995(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  5

  Determinants

  This chapter shows how, when solving systems of n equations in n unknowns, a number, the determinant of the coefficient matrix of the system, arises naturally. To evaluate a determinant directly from its definition can be quite messy so methods to help ease calculation are developed. Despite the (relative) ease with which the Determinants of small matrices are evaluated by hand, Determinants are, nowadays, probably more of theoretical than practical importance. In particular, Cramer’s rule (Exercise 13(b) ) is of little practical value. On the other hand, the criterion given in Theorem 6 , for a matrix to be invertible, is useful in the study of eigenvalues in Chapter 11 .

  Readers who have previously studied Determinants may well find this chapter rather appealing, because it merely confirms, for n × n matrices, what they probably know quite well already in the case of 2 × 2 and 3 × 3 matrices. However, whilst Determinants may have played a more central role in readers’ earlier studies, their role in algebra nowadays is mainly of a theoretical nature: many theorems can conveniently be stated using Determinants – but as a computational tool their usefulness is somewhat limited.

  Nevertheless, let us remind ourselves how Determinants arise in the solving of systems of linear equations. As in Chapter 4 we start with the simplest case again working with letter (rather than actual number) coefficients so that we can better keep track of the various computations which arise.

  Given the simultaneous equations

  (5.1)

  (5.2)

  we may solve them for x 1 and x 2 (cf. equations ax + by = X and cx + dy = Y in Chapter 4

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  eBook - ePub

  A Survey of Modern Algebra

  • Garrett Birkhoff, Saunders Mac Lane(Authors)
  • 2017(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  10 Determinants and Canonical Forms 10.1.  Definition and Elementary Properties of Determinants Over any field each square matrix A has a determinant; though the determinant can be used in the elementary study of the rank of a matrix and in the solution of simultaneous linear equations, its most essential application in matrix theory is to the definition of the characteristic polynomial of a matrix. In this chapter we shall define Determinants, examine their geometric properties, and show the relation of the characteristic polynomial of a matrix A to its characteristic roots (eigenvalues). These concepts will then be applied to the study of canonical forms for matrices under similarity. The formulas for the solution of simultaneous linear equations lead naturally to Determinants. Two linear equations a 1 x + b 1 y = k 1, a 2 x + b 2 y = k 2 have the unique. solution x = (k 1 b 2 − k 2 b 1) / (a 1 b 2 − a 2 b 1), y = (a 1 k 2 − a 2 k 1) / (a 1 b 2 − a 2 b 1), provided a 1 b 2 – a 2 b 1 ≠ 0. The polynomials which appear here in numerator and denominator are known as Determinants, (1) | a 1 b 1 a 2 b 2 | = a 1 b 2 − a 2 b 1, | k 1 b 1 k 2 b 2 | − k 1 b 2 = k 2 b 1. Similarly, one may compute the simultaneous solutions of three linear equations ∑ a i j x j = k j. The denominator of each solution x j turns out. to be (2) | a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 | = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 - a 11 a 23 a 32 - a 12 a 23 a 32 - a 13 a 22 a 31. On the right are six products. Each involves one factor a 1 from the first row, one from the second row, and one from the third. Each column is also represented in every product, so that a term of (2) has the form a 1 _ a 2 _ a 3 with the blanks filled by some permutation of the column indices 1, 2, 3. Of the six possible permutations, the three even permutations I, (123), (132) appear in products with a prefix +, while the odd permutations are associated with a minus sign

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  eBook - ePub

  Circuits, Matrices and Linear Vector Spaces

  • Lawrence P. Huelsman(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  As a compact representation, however, it is a useful one. Frequently we shall omit the brackets and simply refer to the array by the letter alone, i.e., by A. Sometimes, however, for emphasis, it will be helpful to use the containing brackets to indicate that the matrix is square; thus we may write [ A ]. Similarly, a single bracket may be used to indicate a column matrix, as B ]. A horizontal bracket may be used where it is necessary to emphasize that the matrix is a row matrix, as Boldface type will be used whenever a symbol such as A is used to refer to an array of elements without regard to the size of the array. The distinction should be clearly made at this time by the student that a symbol such as A is a label for the matrix, while a ij is a typical element of the matrix, common to row i and column j. This typical element will usually be written as the lowercase of the letter chosen for the label. 2.3 Determinants In studying real or complex variables, considerable attention is given to functions of these variables. In the study of the properties of a matrix, there are also functions to be studied. Of particular interest at this point is a function of the array (more specifically of the elements of the array) called a determinant. To emphasize this point: the matrix is the array, the determinant is a function of the elements of the array. This function only applies to square arrays, i.e., to square matrices, and may be defined by the following steps: Step 1. Form all possible combinations of products of the elements of the array such that each combination contains only one element from each row and each column. For an n × n array there will be n ! of these combinations. Step 2. In each combination arrange the elements according to ascending order of the first subscript. Step 3. Interchange elements in each combination in such a manner that they are now arranged according to the ascending order of the second subscript. Step 4

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  Abstract Algebra with Applications

  Volume 1: Vector Spaces and Groups

  • Karlheinz Spindler(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  7.    Determinants L ET us consider a system of two linear equations for two unknowns, say (⋆) a x + b y = u, c x + d y = υ. To eliminate y from these equations we multiply the first equation by d and the second equation by b and then subtract the two equations thus obtained; this yields (1) (a d − b c) x = d u − b υ. Analogously, multiplying the first equation by c and the second equation by a and then subtracting the two equations thus obtained we find. that (2) (a d − b c) y = a υ − c u. From (1) and (2) we see that there is an alternative between two possible cases. First case: ad − be ≠ 0. In this case the system (*) has a unique solution for any given right-hand side (u, v) T, namely x = d u − b υ a d − b c and y = a υ − c u a d − b c. Second case: ad − be = 0. In this case the system (*) has either no solution at all (if du − bv ≠ 0 or av − cu ≠ 0) or more than one solution (if du − bv = av − cu = 0), depending on the right-hand side (u, v) T. We see that the expression ad − be determines what types of solution sets a system (a b | u c d | υ) can have; it is therefore called the determinant of the coefficient matrix. (7.1) Definition. The determinant of a 2 × 2- matrix is defined as det (a b c d) ≔ | a b c d | ≔ a d − b c. This. formula can be symbolically expressed by the scheme where each of the two diagonals represents the product of the two entries at its ends. With the notion of a determinant at hand we can rewrite the formulas (1) and (2) above as | a b c d | x = | u b υ d | and | a b c d | y = | a u c υ |. L ET us try to find a determinant for (3 × 3)-matrices with analogous properties. Thus let us consider a system of three linear equations for three unknowns, say a 1 x + b 1 y + c 1 z = u 1, a 2 x + b 2 y + c 2 z = u 2, a 3 x + b 3 y + c 3 = u 3. In this system we can try to eliminate y and z by multiplying the three equations by three numbers p, q and r, respectively, and adding the three equations thus obtained; this

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  Precalculus with Limits

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

    (Publisher)

  For example, expanding along the first row yields ∣ A ∣ = a 11 C 11 + a 12 C 12 + . . . + a 1n C 1n . Applying this definition to find a determinant is called expanding by cofactors. 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. 8.4 The Determinant of a Square Matrix 581 GO DIGITAL When expanding by cofactors, you do not need to find cofactors of zero entries, because zero times its cofactor is zero. So, the row (or column) containing the most zeros is usually the best choice for expansion by cofactors. This is demonstrated in the next example. EXAMPLE 4 The Determinant of a 4 × 4 Matrix Find the determinant of A = [ 1 -1 0 3 -2 1 2 4 3 0 0 0 0 2 3 2 ] . Solution Notice that three of the entries in the third column are zeros. So, to eliminate some of the work in the expansion, expand along the third column. ∣ A ∣ = 3(C 13 ) + 0(C 23 ) + 0(C 33 ) + 0(C 43 ) The cofactors C 23 , C 33 , and C 43 have zero coefficients, so the only cofactor you need to find is C 13 . Start by deleting the first row and third column of A to form the determinant that gives the minor M 13 . C 13 = (-1) 1 +3 ∣ -1 0 3 1 2 4 2 3 2 ∣ Delete 1st row and 3rd column. = ∣ -1 0 3 1 2 4 2 3 2 ∣ Simplify. Now, expand by cofactors along the second row. C 13 = 0(-1) 3 ∣ 1 4 2 2 ∣ + 2(-1) 4 ∣ -1 3 2 2 ∣ + 3(-1) 5 ∣ -1 3 1 4 ∣ = 0 + 2(1)(-8) + 3(-1)(-7) = 5 So, ∣ A ∣ = 3C 13 = 3(5) = 15. Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Find the determinant of A = [ 2 2 1 3 6 -2 5 1 -4 3 0 0 2 6 1 -5 ] .

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  Algebra & Trig

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

    (Publisher)

  This is demonstrated in the next example. EXAMPLE 4 The Determinant of a 4 × 4 Matrix Find the determinant of A = [ 1 -1 0 3 -2 1 2 4 3 0 0 0 0 2 3 2 ] . Solution Notice that three of the entries in the third column are zeros. So, to eliminate some of the work in the expansion, expand along the third column. ∣ A ∣ = 3(C 13 ) + 0(C 23 ) + 0(C 33 ) + 0(C 43 ) The cofactors C 23 , C 33 , and C 43 have zero coefficients, so the only cofactor you need to find is C 13 . Start by deleting the first row and third column of A to form the determinant that gives the minor M 13 . C 13 = (-1) 1 +3 ∣ -1 0 3 1 2 4 2 3 2 ∣ Delete 1st row and 3rd column. = ∣ -1 0 3 1 2 4 2 3 2 ∣ Simplify. Now, expand by cofactors along the second row. C 13 = 0(-1) 3 ∣ 1 4 2 2 ∣ + 2(-1) 4 ∣ -1 3 2 2 ∣ + 3(-1) 5 ∣ -1 3 1 4 ∣ = 0 + 2(1)(-8) + 3(-1)(-7) = 5 So, ∣ A ∣ = 3C 13 = 3(5) = 15. Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Find the determinant of A = [ 2 2 1 3 6 -2 5 1 -4 3 0 0 2 6 1 -5 ] . Summarize (Section 10.4) 1. State the definition of the determinant of a 2 × 2 matrix (page 737). For an example of finding the Determinants of 2 × 2 matrices, see Example 1. 2. State the definitions of minors and cofactors of a square matrix (page 739). For an example of finding the minors and cofactors of a square matrix, see Example 2. 3. State the definition of the determinant of a square matrix using expanding by cofactors (page 740). For examples of finding Determinants using expanding by cofactors, see Examples 3 and 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.

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