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$MATLAB C++ Math Library Reference$

326 Chapter 4. Determinants 4.III Other Formulas (This section is optional. Later sections do not depend on this material.) Determinants are a fount of interesting and amusing formulas. Here is one that is often seen in calculus classes and used to compute determinants by hand. 4.III.1 Laplace’s Expansion 1.1 Example In this permutation expansion � �t1,1 � �t2,1 � �t3,1 t1,2 t2,2 t3,2 � � � � � t1,3� � � �1 0 0� � � �1 0 0� � t2,3� � = t1,1t2,2t3,3 � �0 1 0� � + t1,1t2,3t3,2 � �0 0 1� � t3,3� �0 0 1� �0 1 0� � � � � �0 1 0� � � �0 1 + t1,2t2,1t3,3 � �1 0 0� � + t1,2t2,3t3,1 � �0 0 �0 0 1� �1 0 � � � � �0 0 1� � � �0 0 + t1,3t2,1t3,2 � �1 0 0� � + t1,3t2,2t3,1 � �0 1 �0 1 0� �1 0 � 0� � 1� � 0� � 1� � 0� � 0� we can, for instance, factor out the entries from the first row ⎡ � � � � ⎤ � �1 0 0� � � �1 0 0� � = t1,1 · ⎣t2,2t3,3 � �0 1 0� � + t2,3t3,2 � �0 0 1�⎦ � �0 0 1� �0 1 0� ⎡ � � � � ⎤ � �0 1 0� � � �0 1 0� � + t1,2 · ⎣t2,1t3,3 � �1 0 0� � + t2,3t3,1 � �0 0 1�⎦ � �0 0 1� �1 0 0� ⎡ � � � � ⎤ � �0 0 1� � � �0 0 1� � + t1,3 · ⎣t2,1t3,2 � �1 0 0� � + t2,2t3,1 � �0 1 0�⎦ � �0 1 0� �1 0 0� and swap rows in the permutation matrices to get this. ⎡ � � � � ⎤ � �1 0 0� � � �1 0 0� � = t1,1 · ⎣t2,2t3,3 � �0 1 0� � + t2,3t3,2 � �0 0 1�⎦ � �0 0 1� �0 1 0� ⎡ � � � � ⎤ � �1 0 0� � � �1 0 0� � − t1,2 · ⎣t2,1t3,3 � �0 1 0� � + t2,3t3,1 � �0 0 1�⎦ � �0 0 1� �0 1 0� ⎡ � � � � ⎤ � �1 0 0� � � �1 0 0� � + t1,3 · ⎣t2,1t3,2 � �0 1 0� � + t2,2t3,1 � �0 0 1�⎦ � �0 0 1� �0 1 0�
Section III. Other Formulas 327 The point of the swapping (one swap to each of the permutation matrices on the second line and two swaps to each on the third line) is that the three lines simplify to three terms. � � = t1,1 · � � t2,2 � t2,3� � � − t1,2 � � · � � t2,1 � t2,3� � � + t1,3 � � · � � t2,1 � t2,2� � � t3,2 t3,3 t3,1 t3,3 t3,1 t3,2 The formula given in Theorem 1.5, which generalizes this example, is a recurrence — the determinant is expressed as a combination of determinants. This formula isn’t circular because, as here, the determinant is expressed in terms of determinants of matrices of smaller size. 1.2 Definition For any n×n matrix T , the (n − 1)×(n − 1) matrix formed by deleting row i and column j of T is the i, j minor of T .Thei, j cofactor Ti,j of T is (−1) i+j times the determinant of the i, j minor of T . 1.3 Example The 1, 2 cofactor of the matrix from Example 1.1 is the negative of the second 2×2 determinant. � � T1,2 = −1 · � � t2,1 � t2,3� � � 1.4 Example Where t3,1 t3,3 ⎛ 1 2 ⎞ 3 T = ⎝4 5 6⎠ 7 8 9 these are the 1, 2 and 2, 2 cofactors. T1,2 =(−1) 1+2 � � · �4 �7 � 6� � 9� =6 T2,2 =(−1) 2+2 � � · �1 �7 � 3� � 9� = −12 1.5 Theorem (Laplace Expansion of Determinants) Where T is an n×n matrix, the determinant can be found by expanding by cofactors on row i or column j. |T | = ti,1 · Ti,1 + ti,2 · Ti,2 + ···+ ti,n · Ti,n = t1,j · T1,j + t2,j · T2,j + ···+ tn,j · Tn,j Proof. Exercise 27. QED 1.6 Example We can compute the determinant � � �1 |T | = � �4 �7 2 5 8 � 3� � 6� � 9�
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� � � �1 2� � �3 1�
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Preface In most mathematics program
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Contents 1 Linear Systems 1 1.I Sol
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5.II.1 Definition and Examples ....
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2 Chapter 1. Linear Systems To fini
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4 Chapter 1. Linear Systems Conside
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22 Chapter 1. Linear Systems Studyi
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26 Chapter 1. Linear Systems shows
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28 Chapter 1. Linear Systems 3.13 E
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30 Chapter 1. Linear Systems (a) 2x
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32 Chapter 1. Linear Systems 1.II L
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34 Chapter 1. Linear Systems positi
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36 Chapter 1. Linear Systems In R3
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38 Chapter 1. Linear Systems � 1.
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40 Chapter 1. Linear Systems Notice
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42 Chapter 1. Linear Systems 2.7 De
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44 Chapter 1. Linear Systems 2.25 P
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56 Chapter 1. Linear Systems First,
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80 Chapter 2. Vector Spaces 2.I Def
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82 Chapter 2. Vector Spaces For the
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86 Chapter 2. Vector Spaces 1.12 Ex
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90 Chapter 2. Vector Spaces (d) The
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92 Chapter 2. Vector Spaces 2.2 Exa
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94 Chapter 2. Vector Spaces second.
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96 Chapter 2. Vector Spaces Proof.
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98 Chapter 2. Vector Spaces The sub
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100 Chapter 2. Vector Spaces S. Mem
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102 Chapter 2. Vector Spaces 2.II L
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104 Chapter 2. Vector Spaces 1.5 Ex
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106 Chapter 2. Vector Spaces Lookin
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108 Chapter 2. Vector Spaces Proof.
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110 Chapter 2. Vector Spaces (a) {2
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112 Chapter 2. Vector Spaces � 1.
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128 Chapter 2. Vector Spaces Anothe
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130 Chapter 2. Vector Spaces (a)
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136 Chapter 2. Vector Spaces 4.12 E
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146 Chapter 2. Vector Spaces (d) Pl
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148 Chapter 2. Vector Spaces howeve
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150 Chapter 2. Vector Spaces positi
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154 Chapter 2. Vector Spaces The cl
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160 Chapter 3. Maps Between Spaces
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Section IV. Jordan Form 379 5.IV Jo
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Topic: Linear Recurrences 407 And,
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Appendix Introduction Mathematics i
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A-3 ‘if a number is divisible by
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Techniques of Proof A-5 Induction.
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A-7 The Principle of Extensionality
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Bibliography [Abbot] Edwin Abbott,
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[Feller] William Feller, An Introdu
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[Programmer’s Ref.] Microsoft Pro
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congruent plane figures, 286 contra
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Gauss’ method, 3 Gauss-Jordan, 46
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Linear Algebra

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