Chapter 2 Determinant

1. Chapter 2Determinant • Objective: To introduce the notion of determinant, and some of its properties as well as applications.

2. Several applications of determinant • Test for singularity of a matrix instead of by definition. • Find the area of a parallelogram generated by two vectors. • Find the volume of a parallelopipe spanned by three vectors. • Solve Ax=b by Cramer’s rule.

3. Introduction to Determinant (to determine the singularity of a matrix) Consider . If we define det(A)=a, then A is nonsingular.

4. Case2 2×2 Matrices • Let • Suppose , then A If we define then A is nonsingular

5. Case2 2×2 Matrices (cont.) • Suppose but , then A and Thus A is nonsingular • Suppose A is singular & det(A)=0. • To summarize, A is nonsingular

6. Case3 3×3 Matrices • Let -Suppose , A

7. Case3 3×3 Matrices (cont.) • From 2x2 case, A I Then A is nonsingular define

8. Easily Shown for Cases that AI

9. Recall • For , where

10. Recall (cont.) • For , where , ,

11. Generalization Definition: Let , , and let • the matrix obtained from A by deleting the row & column containing • The is called the minor of • The cofactor of is denoted as

12. Definition: The determinant of is defined as , if n=1 , if n>1 Note: det is a function from to .

13. Theroem2.1.1:Let , Hint: By induction or sign-type definition.

14. Theroem2.1.2: Let ,and Pf: By induction, n=1,ok! Suppose the theorem is true for n=k. If n=k+1, By induction The result then follows.

15. Theroem2.1.3: Let be a triangular matrix. Then Hint:expansion for lst row or column and induction on n. Theroem2.1.4: (i)If A has a row or column consisting entirely of zeros, then (ii)If A has two identical rows or columns, then Hint for (ii): By mathematical induction.

16. §2-2 Properties of Determinants Note that For example, , Question: Is

17. Lemma2.2.1: Let , then

18. Proof of Lemma2.2.1 Pf: Case for i=j follows directly from the definition of determinant. For , define to be the matrix obtained from A by replacing the jth row of A by ith row of A. (Then has two identical rows) expansion along jth row

19. Proof of Lemma2.2.1(cont.) jth row

20. Note that • by Th. 2.1.3 • by Th. 2.1.3 • 先對非交換列展開 數學歸納法

21. Lemma 2.2.1

23. Thus, we have If E is an elementary matrix In fact, det(AE)=det(A)det(E) Question:

24. Theorem2.2.2: is singular Pf:Transform A to its row echelor from as If A is singular If A is nonsingular The result then follows.

25. Theorem2.2.3:Let .Then Pf: If B is singular AB is singular If B is nonsingular

26. §2-3 Cramer’s Rule • Objective: Use determinant to compute and solve Ax=b.

27. The Adjoint of a Matrix Def: Let .The adjoint of A is defined to be where are cofactor of

28. By Lemma2.2.1, we have If A is nonsingular, det(A) is a nonzero scalar

29. Example 1 (P.116) For a 2×2 matrix : If A is nonsingular , then

30. Example 2 (P.116) Q: Let , compute adj A and A-1. Sol:

31. Theorem2.3.1:(Cramer’s Rule) Let be nonsingular and . Denote the matrix obtained by replacing the ith column of A by .Then the unique sol. of is Pf:

32. Example 3 (P.117) Q: Use Cramer’s rule to Solve

33. Example 3 (cont.) Sol:

34. Let .Then volume of the parallelopipe spanned by and is • Let .Then the area of the parallelogram spanned by and is

35. Application 1: Coded Message (P.118) For example, the message Send Money might be coded as 5, 8, 10, 21, 7, 2, 10, 8, 3 here the S is represented by a “5”, the E is represented by a “8”, and so on.

36. Application 1: Coded Message (cont.) If A is a matrix whose entries are all integers and whose determinants is ± 1, then, since , the entries of A-1 will be integers. Let

37. Application 1: Coded Message (cont.) We can decode it by multiplying by A-1 We can construct A by applying a sequence of row operations on identity matrix. Note: A-1 AB(encoding Message)