Linear Algebra

1. Linear Algebra Chapter 3 Determinants （行列式）

2. Topics • Preliminaries（前言） • Definition（定義） • Properties of Determinants（行列式的性質） • Cofactor Expansion（余子式展開） • Inverse of a Matrix（矩陣的逆） • Other Applications of Determinants（行列式的應用）

3. Topics • Preliminaries • Definition • Properties of Determinants • Cofactor Expansion • Inverse of a Matrix • Other Applications of Determinants

4. Preliminaries • Consider the system of linear equations which can be written in matrix form as Can show that this system has a solution if and only if ad-bc ≠ 0. The number ad-bc, which also appears in the solution of the above equations, is called the determinant of the coefficient matrix（係數矩陣的行列式） • Want to generalize this to a square matrix of any size

5. Preliminaries Uses of Determinants（行列式的應用） • Condition for singularity is det (A) = 0. Eigenvalues（特征值）of A are those values lthat make the matrix A – lI singular. Condition det (A – lI) = 0 gives a polynomial（多项式）whose roots are the eigenvalues。 • Explicit formulas for solutions of long-standing problems: • solution of linear systems • computation of matrix inverse • Explicit formula for A–1 is of limited computational utility, but it does show how A–1 depends on the n2 entries of A and how it varies when those entries vary。

6. Topics • Preliminaries • Definition • Properties of Determinants • Cofactor Expansion • Inverse of a Matrix • Other Applications of Determinants

7. Definition • Defn - Let S = { 1, 2, …, n } be the integers 1 through n, arranged in ascending order. A rearrangement j1 j2 … jn of the elements of S is called a permutation（排列）of S. A permutation of S is a one to one mapping (映射)of S onto itself. • The number of permutations of S = { 1, 2, …, n } is n! • The set of all permutations of the elements of S is denoted by Sn • It will be helpful to classify permutations as even or odd

8. inversion inversion inversion Definition • Defn - A permutation j1 j2 … jn of S is said to have an inversion（逆序）if a larger integer, jr, precedes a smaller one, js. Example In the permutation 32514 in S5， 3 2 5 1 4

9. Total number of inversions(逆序数) 分别计算出排列中每个元素前面比它大的数码个数，即算出排列中每个元素的逆序数，则所有元素的逆序数之总和即为所求排列的逆序数. Example 3 2 5 1 4 1 3 So the total number of inversions is 0+1+0+3+1=5.

10. Definition • Example - Let S = { 1, 2, 3 }. Then S3= { 123, 132, 213, 231, 312, 321 } 123- 0 inversions 132- 1 inversion 213 - 1 inversion 231- 2 inversions 312- 2 inversions 321- 3 inversions

11. Definition • Defn - A permutation is called even（偶排列）if the total number of inversions(逆序数) is even • Defn - A permutation is called odd（奇排列）if the total number of inversions（逆序数）is odd • Example • 4312 contains 5 inversions and is an odd permutation. • 1423 contains 2 inversions and is an even permutation

12. 练习：计算下列排列的逆序数，并讨论它们的奇偶性.练习：计算下列排列的逆序数，并讨论它们的奇偶性. 解 （1） 此排列为偶排列.

13. 当 时为偶排列； 当 时为奇排列. 解

14. Definition • Defn - Let A= [ aij] be an n×n matrix. The determinant of A，denoted by det(A),is defined as where the summation is over all permutations j1 j2 … jn of S = { 1, 2, …, n }. The sign is + if j1 j2 … jn is an even permutation and is- if j1 j2 … jn is an odd permutation

15. Definition Comments • det (A) is a function from the set of n×n matrices into the real numbers • Each term of det (A) has row subscripts in natural order and column subscripts in the order j1 j2 … jn . Thus each term is a product of n elements of A, with exactly one entry from each row of A and exactly one entry from each column of A. • det (A) is also written

16. Definition Examples • Let A = [ a11 ], a 1×1 matrix, then det (A) =a11 • Let det (A) =a11 a22-a12 a21

17. Example if ， Definition • One trick(诀窍) for the determinants Diagonal Rule(对角线法则) Main diagonal （主对角线） Minor diagonal（副对角线） then

18. Definition Examples • Let det (A) =a11 a22 a33 + a12 a23 a31 + a13 a21 a32 -a11 a23 a32 - a12 a21 a33 -a13 a22 a31

19. Two tricks(诀窍) for the determinants (1) Rule of Sarrus（沙路法）

20. (2) Diangonal rule（对角线法则）

21. Example

22. Example

23. Exercise

24. Warning: The method used for computing the 2 ×2 determinants and the 3 ×3 determinants do not apply for n≥4.

26. Topics • Preliminaries • Definition • Properties of Determinants • Cofactor Expansion • Inverse of a Matrix • Other Applications of Determinants • Computing Determinants

27. Properties of Determinants • Theorem3.1 - Let A be a square matrix. Then det (A) = det (AT) Comment - Since det (A) = det (AT), the properties of determinants with respect to row manipulations are true also for the corresponding column manipulations

28. Properties of Determinants • Theorem3.2 - If matrix B is obtained from matrix A by interchanging two rows (columns) of A, then det (B) =- det (A) • Theorem3.3 - If two rows (columns) of A are equal, then det (A) = 0.

29. Properties of Determinants • Theorem3.4 - If a row (column) of A consists entirely of zeros, then det (A) = 0. Example

30. Properties of Determinants • Theorem3.5 - If B is obtained from A by multiplying a row (column) of A by a real number k, then det ( B ) =k det ( A ).

31. Example: Let A= (a ij)be a n×n matrix ,and k be a real number Proof |kA|= kn|A|.

32. Properties of Determinants • Theorem3.6 - If B= [ bij ] is obtained from A= [ aij ] by adding to each element of rth row (column) of A, k times the corresponding element of the sth row (column), r ≠ s, of A, then det(B) = det(A)

33. Properties of Determinants Comment • The previous three theorems describe what happens to the determinant of a matrix A when the following elementary row (column) operations are performed on it • Type I - Interchange two rows (columns) of A • Type II - Multiply row (column) i of A by c ≠ 0. (Theorem also covers the case c= 0) • Type III - Add c times row (column) i of A to row (column) j of A, i ≠ j

34. Properties of Determinants • Theorem3.7 - Let A= [ aij ] be an upper (lower) triangular matrix, then det(A) =a11a22 … ann. That is, the determinant of a triangular matrix is just the product of the elements on the main diagonal.

37. Example：Compute the determinant =9

38. Exercise：Compute

39. Exercises.

41. Properties of Determinants Comment • Each elementary row operation may be accomplished by multiplying A by the appropriate elementary matrix, which is obtained by performing the operation on the identity matrix I. By the preceding theorem, det(I) = 1. Let EI, EII and EIII be elementary matrices of Type I, Type II and Type III respectively. Then, by the theorems proved recently det(EI) = – 1 det(EII) =c det(EIII) = 1

42. Properties of Determinants • Lemma3.1 - Let E be an elementary matrix. Then det(EA) = det(E) det(A) and det(AE) = det(A) det(E)

43. Properties of Determinants • Theorem3.8 - Let A be a n×n matrix. Then A is nonsingular if and only if det(A) ≠ 0

44. Properties of Determinants • Corollary3.1 - If A is an n×n matrix, then Ax=0 has a nontrivial solution if and only if det(A) = 0.

45. Properties of Determinants • Theorem3.9 - If A and B are n × n matrices, then det(AB) = det(A) det(B)

46. Example Properties of Determinants • Theorem - If A is nonsingular, then det(A–1) = 1/det(A) • Proof - Let In be the nxn identity. Then AA–1=In and det(In) = 1. By the preceding theorem, det(A) det(A–1) = 1, so det(A–1) = 1/det(A)