Linear Algebra Chapter 2 : DETERMINANT Instructor Dr. Dang Van Vinh (6/2006)

1. Math Dept, Faculty of Applied Science, HCM University of Technology------------------------------------------------------------------------------------- Linear Algebra Chapter 2: DETERMINANT • Instructor Dr. Dang Van Vinh (6/2006)

2. CONTENTS---------------------------------------------------------------------------------------------------------------------------CONTENTS--------------------------------------------------------------------------------------------------------------------------- I – The Determinant of Matrix II – Properties of Determinant III – Laplace’s Expansion

3. Let be a square matrix. Determinant of A is denoted by det Let denote the submatrix formed by deleting the i th row and j th column of A; Definition of (i,j)- cofactor (i,j)- cofactor: I. The Determinant of Matrix---------------------------------------------------------------------------------------------------------------------------

4. Recursive definition of determinant a) k =1: b) k =2: c) k =3: d) k =n: I. The Determinant of Matrix--------------------------------------------------------------------------------------------------------------------------- ...............

5. Example Compute det (A), where Solution I. The Determinant of Matrix---------------------------------------------------------------------------------------------------------------------------

6. The determinant of an matrix A may be computed by a cofactor expansion along any row or down any column II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

7. Example Compute det (A), where Solution. We expand across the 3th row II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

8. Example Compute det (A), where II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

9. Solution II. Properties of Determinant --------------------------------------------------------------------------------------------------------------------------- We expand down the 2th column

10. If A is a triangular matrix, then det(A) is the product of the entries on the main diagonal of A. Example II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

11. Using Row Operation for Calculating Determinant Row operations II. Properties of Determinant --------------------------------------------------------------------------------------------------------------------------- a. If a multiple of one row of A is added to another to produced a matrix B, then det(B) = det(A). b. If two row of A interchanged to produced B, then det(B) = - det(A). c. If one row of A is multiplied by scalar k to produced a B, then det(B) =k det(A). Triangular matrix B A

12. Using Row Operation for Calculating Determinant 1. If then 2. If then 3. If then II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

13. Example Find the determinant of a matrix A, using elementary operations II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

14. Solution II. Properties of Determinant --------------------------------------------------------------------------------------------------------------------------- Expand according to the first column

15. The Formula of Calculating Determinant II. Properties of Determinant --------------------------------------------------------------------------------------------------------------------------- Step 1. Select one column (or one row) of the matrix Step 2. Choose one nonzero element of the selected column (or selected row). Using row ( or column) operations to eliminate all others elements except selected. Step3. Expand the determinant according to the selected row ( or column)

16. Example Find the determinant of a matrix A, using elementary operations II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

17. Solution Expand according to the fourth column II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

18. A cofactor expansion requires over n! multiplications. If a supercomputer could make one trillion multiplications per second, it would have to run for over 500.000 years to compute a 25x25 determinant by cofactor expansion (required 25! is approximately 1.5x1025 operations). Most computer programs that compute det (A) using a row operations. The row operations requires (n3+2n-3)/3 multiplications and divisions. Any modern microcomputer can calculate a 25x25 determinant in a fraction of a second, since less than 5300 such operations are required. II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

19. If A is an nxn matrix, then det (AT) = det (A) det(AB) = det(A) det(B) Warning: det(A+B) is not equal to det(A) + det(B) in general If a matrix A has a zero row, then det (A) = 0 If a matrix A has two identical rows, then det (A) = 0 II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

20. Theorem A square matrix A is invertible if and only if det(A) 0. Proof det(AA-1) = det (I) det(A).det(A-1) = 1 det(A) 0 Suppose that det(A) 0. Then , where II. Properties of Determinant --------------------------------------------------------------------------------------------------------------------------- Let A be an invertible nxn matrix. There exists an inverse A-1, such that AA-1 = I. It follows that

21. II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

22. Determinant Formula for A-1 Let A be an invertible nxn matrix. Then , where II. Properties of Determinant ---------------------------------------------------------------------------------------------------------------------------

23. Exp. Find the inverse of the matrix A is invertible II. Properties of Determinant --------------------------------------------------------------------------------------------------------------------------- Solution. The nine cofactors are

24. Properties of an invertible matrix 1. 2. If A is invertible, then II. Properties of Determinant --------------------------------------------------------------------------------------------------------------------------- Proof.

25. III. Laplace’s Expansion ----------------------------------------------------------------------------- Definition of a sub-determinant of order k The sub-determinant of order k, denoted by , is determinant of order k corresponding to the matrix formed by the elements of matrix A lying at the intersection of k rows labeled i1, i2, …, ik and k columns labeled j1, j2, …, jk Definition of k-Minor The k-minor of the sub determinant of order k is determinant of order n - k corresponding to the matrix obtained from A by deleting k rows labeled i1, i2, …, ik and k columns labeled j1, j2, …, jk Suppose k is any natural number smaller than n and i1, i2, …, ik and j1, j2, …, jk are arbitrary numbers satisfying the conditions

26. The quantity is called a k – cofactor of Theorem (Laplace’s Expansion) For any natural number k smaller than n and for any fixed numbers of rows i1, i2, …, ik such that , the following formula holds true This formula is called the expansion of the determinant according to k rows i1, i2, …, ik. The summation in this formula is carried out by all possible values of the indeces j1, j2, …, jk satisfying the conditions III. Laplace’s Expansion -----------------------------------------------------------------------------

27. Example Find the determinant of a matrix A, using Laplace’s Expansion. III. Laplace’s Expansion -----------------------------------------------------------------------------

28. Solution There are kxk submatrices of the first type, but only one nonzero. III. Laplace’s Expansion ----------------------------------------------------------------------------- Select k = 2 and select 2 rows: the second and the fourth rows

29. Example 1 Calculate det(A), where

30. Example 2 Calculate det(A), where

31. Example 3 Which one of the following statements is true? a) The degree of polynomial f(x) is 5. b) The degree of polynomial f(x) is 4. c) The degree of polynomial f(x) is 3. d) The others statements are false.

32. Example 4 Calculate the determinant of the following matrix

33. Example 5 Calculate the determinant

34. Example 6 Solve the equation, where a, b, c are real numbers.

35. Example 7 Solve the equation

36. Example 8 Find the determinant

37. Example 9 Calculate

38. Solution Expand along the first row

40. Example 10 Find

41. Example 11 Compute

42. Example 12 Solve the following equation in C

43. Example 13 Calculate det(A) using Laplace’s Expansion, where

44. Example 14 Find an inverse of the following matrix A, using the Determinant’s formula

45. Example 15 Find an inverse of the following matrix A, using the Determinant’s formula

46. Example 16 Find all m such that a matrix A is invertible.

47. Example 17 Find all m such that a matrix A is invertible.

48. Example 18 Let . 1) Calculate det (A-1). 2) Calculate det (5A)-1. 3) Calculate det (PA).

49. Example 19 1) Find det (4AB)-1. 2) Find det (PAB). Let