1. CHAPTER 7 Determinant s

2. Outline - Permutation - Definition of the Determinant - Properties of Determinants - Evaluation of Determinants by Elementary Row and Column Operation - Cofactor Expansions - Determinants of Triangular Matrices

3. - A Determinant Formula for a Matrix Inverse - Cramer's Rule - The Matrix Tree Theorem

4. Mathematics for Computer Engineering 4 Permutations A permutation of order n is an arrangement of the integers 1,...,n in any order.

5. Mathematics for Computer Engineering 5 Example Suppose p is a permutation that reorders the integers 1,...,6 as 3,1,4,5,2,6 P(1) = 3, p(2) = 1, p(3) = 4, p(4) = 5, p(5) = 2, p(6) = 6 with p(j) the number the permutation has put in place j.

6. Mathematics for Computer Engineering 6 There are n! = 1 * 2 * 3 * 4 *... * n permutations on 1,...,n. Example For n = 3 there are 6 permutations on 1,2,3. For n = 4 there are 24 permutations on 1,2,3,4.

7. Mathematics for Computer Engineering 7 A permutation is characterized as even or odd, according to a rule we will now illustrate. Consider the permutation 2,5,1,4,3 on the integers 1,...,5. for each number k in the list, count the number of integers to its right that are smaller than k.

8. Mathematics for Computer Engineering 8 k number of integers smaller than k to the right of k 21 53 10 41 30 Sum the integers in the right column to get 5, which is odd. We therefore call this permutation odd.

9. Mathematics for Computer Engineering 9 Consider the permutation 2,1,5,4,3 on the integers 1,...,5. for each number k in the list, count the number of integers to its right that are smaller than k.

10. Mathematics for Computer Engineering 10 k number of integers smaller than k to the right of k 21 10 52 41 30 Sum the integers in the right column to get 4, which is even. We therefore call this permutation even.

11. Mathematics for Computer Engineering 11 If p is a permutation, let

12. Mathematics for Computer Engineering 12 Definition of the Determinant DEFINITION 7.1 Each product in the sum is multiplied by 1 if the permutation is even and by -1 if p is odd.

13. Mathematics for Computer Engineering 13 Example n=2 p:1,2 even permutation q:2,1 odd permutation

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15. Mathematics for Computer Engineering 15 Example n=3

16. Mathematics for Computer Engineering 16 The permutations of 1,2,3 are p 1 : 1,2,3 even permutation p 2 : 1,3,2 odd permutation p 3 : 2,1,3odd permutation p 4 : 2,3,1even permutation p 5 : 3,1,2even permutation p 6 : 3,2,1odd permutation

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18. Mathematics for Computer Engineering 18 Properties of Determinants det(A) = |A| This should not be confused with absolute value.

19. Mathematics for Computer Engineering 19 Theorem 7.1 If A has a zero row, then |A| =0.

20. Mathematics for Computer Engineering 20 Theorem 7.2 Let B be formed from A by multiplying row k by a scalar α. Then | B | = α| A |

21. Mathematics for Computer Engineering 21 Theorem 7.3 Let B be formed from A by interchanging two rows. Then | A | = -| B |

22. Mathematics for Computer Engineering 22 COROLLARY 7.1 If two rows of A are the same, then | A | = 0

23. Mathematics for Computer Engineering 23 COROLLARY 7.2 If for some scalar α, row k of A is α time row i, Then | A | = 0

24. Mathematics for Computer Engineering 24 THEOREM 7.4 Let A and B be n*n matrices. Then | AB | = | A || B |

25. Mathematics for Computer Engineering 25 THEOREM 7.5 Suppose each element of row k of A is written as a sum α kj +β kj. Form two matrices from A. The first, A 1, is identical to A expect the elements of row k are α kj. The second, A 2, is identical to A expect the elements of row k are β kj. Then | A | = | A 1 |+| A 2 |

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28. Mathematics for Computer Engineering 28 COROLLARY 7.3 Let B be formed from A by adding γ times row i to row k. Then | B | = | A |

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30. Mathematics for Computer Engineering 30 The effect of elementary row operations on a determinant Type I operation - interchange of two rows. This changes the sign of determinant. Type II operation - multiplication of a row by a scalar α. This multiplies the determinant by α. Type III operation – addition of a scalar multiple of one row to another row. This does not change the determinant.

31. Mathematics for Computer Engineering 31 THEOREM 7.6 | A | = | A t |

32. Mathematics for Computer Engineering 32 Evaluation of Determinants by Elementary Row and Column Operation If a row or column of an n*n matrix A has all zero elements expect possibly for a ij in row i and column j, then the determinant of A is (-1) i+j a ij times the determinant of the (n-1)*(n-1) matrix obtained by deleting row i and column j from A.

33. Mathematics for Computer Engineering 33 THEOREM 7.7 Row Version

34. Mathematics for Computer Engineering 34 Column Version

35. Mathematics for Computer Engineering 35 Example 7.1

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41. Mathematics for Computer Engineering 41 Example 7.2

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45. Mathematics for Computer Engineering 45 Cofactor Expansions

46. Mathematics for Computer Engineering 46 Definition 7.2

47. Mathematics for Computer Engineering 47 THEOREM 7.8 Cofactor Expansion by a row If A is n*n, then for any integer i with 1 ≤ i ≤ n

48. Mathematics for Computer Engineering 48 Example 7.3

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50. Mathematics for Computer Engineering 50 THEOREM 7.9 Cofactor Expansion by a Column Let A be an n*n. Then for any integer j with 1 ≤ j ≤ n

51. Mathematics for Computer Engineering 51 Example 7.4

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53. Mathematics for Computer Engineering 53 Determinants of Triangular Matrices A upper triangular matrix that all the elements below the main diagonal are zero. The main diagonal of a square matrix A consists of the elements a 11, a 22, a 33,...,a nn

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57. Mathematics for Computer Engineering 57 The determinant of an upper triangular matrix is the product of its main diagonal elements. The same conclusion holds for lower triangular matrices (all elements above the main diagonal are zero)

58. Mathematics for Computer Engineering 58 Example 7.5

59. Mathematics for Computer Engineering 59 A Determinant of Formula for a Matrix Inverse

60. Mathematics for Computer Engineering 60 THEOREM 7.10 Let A be an n*n matrix. Then A is nonsingular if and only if |A| ≠ 0.

61. Mathematics for Computer Engineering 61 THEOREM 7.11 Let A be an n*n nonsingular matrix. Define an n*n B by putting Then, B = A -1

62. Mathematics for Computer Engineering 62 Example 7.6

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65. Mathematics for Computer Engineering 65 Cramer's Rule Cramer's rule is a determinant formula for solving a system of equations AX = B when A is n*n and nonsingular. X = A -1 B

66. Mathematics for Computer Engineering 66 THEOREM 7.12 Let A be a nonsingular n*n matrix of numbers. Then the unique solution of AX=B is

67. Mathematics for Computer Engineering 67 where and A(k;B) is the matrix obtained from A by replacing column k of A by B.

68. Mathematics for Computer Engineering 68 Example 7.7 Solve the system

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70. Mathematics for Computer Engineering 70 The Matrix Tree Theorem G.R. Kirchhoff published the electrical circuit laws of matrix tree theorem in 1847. A spanning tree is a collection of lines in the graph forming no closed loops, containing a path between any two points of the graph.

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72. Mathematics for Computer Engineering 72 Kirchhoff derived a relationship between determinants and the number of labeled spanning tree in a graph.

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74. Mathematics for Computer Engineering 74 THEOREM 7.13 Matrix Tree Theorem Let G be a graph with vertices labeled v 1,...,v n. Form an n*n matrix T=[t ij ] as follows. If i=j, then t ii is the number of lines to v i in the graph. If i ≠ j, then t ij = 0 if there is no line between v i and v j in G, and t ij =-1 if there is such a line. Then, all cofactors of T are equal, and their common value is the number of spanning trees in G.

75. Mathematics for Computer Engineering 75 Example 7.8

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