Chapter 2B Determinants

1. Chapter 2BDeterminants When we look at a particular square matrix, the question of whether it is nonsingular is one of the first things that we ask. This chapter develops a formula to determine this. • 2B.1 The Determinant and Evaluation of a Matrix • 2B.2 Properties of Determinants • 2B.3 Eigenvalues and Application of Determinants • 2B.4 Geometry of Determinants: Determinants as Size Functions

2. 2B.1 The Determinant of a Matrix • The determinant of a 2 × 2 matrix: • Note:

3. Minor of the entry : • Cofactor of :

4. Ex: • Notes: Sign pattern for cofactors

5. Thm 3B.1: (Expansion by cofactors) Let A isasquare matrix of order n, then the determinant of A is given by (Cofactor expansion along the i-th row, i=1, 2,…, n) or (Cofactor expansion along the j-th column, j=1, 2,…, n )

6. Ex: The determinant of a matrix of order 3

7. Ex 5: (The determinant of a matrix of order 3) Sol:

8. –4 0 6 • Ex : 0 –12 16 • The determinant of a matrix of order 3:

9. Upper triangular matrix: All the entries below the main diagonal are zeros. • Lower triangular matrix: All the entries above the main diagonal are zeros. • Diagonal matrix: All the entries above and below the main diagonal are zeros. diagonal upper triangular lower triangular

10. Theorem 2B.2: Determinant of a Triangular Matrix If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is At this moment, our primary way to decide whether a matrix is singular is to do Gaussian reduction and then check whether the diagonal of resulting echelon form matrix has any zeroes. We will look for a family of functions with the property of being unaffected by row operations and with the property that a determinant of an echelon form matrix is the product of its diagonal entries.

11. (b) Ex:Find the determinants of the following triangular matrices. (a) Sol: |A| = (2)(–2)(1)(3) = –12 (a) |B| = (–1)(3)(2)(4)(–2) = 48 (b)

12. Keywords in This Section: • determinant : 行列式 • minor : 子行列式 • cofactor : 餘因子 • expansion by cofactors : 餘因子展開 • upper triangular matrix: 上三角矩陣 • lower triangular matrix: 下三角矩陣 • diagonal matrix: 對角矩陣

13. 2B.2 Evaluation of a determinant using elementary operations • Theorem 2B.3: Elementary row operations and determinants Let A and B be square matrices,

14. Ex:

15. Sol: Note: A row-echelon form of a square matrix is always upper triangular. • Ex: Evaluation a determinant using elementary row operations

17. Theorem 2B.4: Conditions that yield a zero determinant If A is a square matrix and any of the following conditions is true, then det (A) = 0. (a) An entire row (or an entire column) consists of zeros. (b) Two rows (or two columns) are equal. (c) One row (or column) is a multiple of another row (or column). The theorem states that : a matrix with two identical rows or two linear dependent rows has a determinant of zero. A matrix with a zero row has a determinant of zero. Note that a matrix is nonsingular if and only if its determinant is nonzero and the determinant of an echelon form matrix is the product down its diagonal. This theorem provides a way to compute the value of a determinant function on a matrix:Do Gaussian reduction, keeping track of any changes of sign caused by row swaps and any scalars that are factored out, and then finish by multiplying down the diagonal of the echelon form result.

18. Note:

19. Ex: (Evaluating a determinant) Sol:

21. 2B.2 Properties of Determinants • Notes: • Theorem 2B.5: Determinant of a matrix product det (AB) = det (A) det (B) (1) det (EA) = det (E) det (A) (2) (3)

22. Ex: (The determinant of a matrix product) Find |A|, |B|, and |AB| Sol:

24. Ex: • Theorem 2B.6: Determinant of a scalar multiple of a matrix If A is an n × n matrix and c is a scalar, then det (cA) = cn det (A) Find |A|. Sol:

25. Thm 2B.7: Determinant of an invertible matrix • Ex: (Classifying square matrices as singular or nonsingular) A square matrix A is invertible (nonsingular) if and only if det (A)  0 Sol: A has no inverse (it is singular). B has inverse (it is nonsingular).

26. Ex: • Thm 2B.8: Determinant of an inverse matrix • Thm 2B.9: Determinant of a transpose (a) (b) Sol:

27. If A is an n × n matrix, then the following statements are equivalent. • Equivalent conditions for a nonsingular matrix: (1) A is invertible. (2) Ax = b has a unique solution for every n × 1 matrix b. (3) Ax = 0 has only the trivial solution of zero column vector. (4) A is row-equivalent to In (5) A can be written as the product of elementary matrices. (6) det (A)  0

28. Ex: Which of the following system has a unique solution? (a) Sol: This system does not have a unique solution.

29. This system has a unique solution. Sol: (b)

30. Eigenvalue and eigenvector: A：an nn matrix ：a scalar x： a n1nonzero column matrix Eigenvalue Eigenvector (The fundamental equation for the eigenvalue problem) 2B.3 Introduction to Eigenvalues • Eigenvalue problem: If A is an nn matrix, do there exist nonzero n1 matrices x such that Ax is a scalar multiple of x？

31. Eigenvalue Eigenvalue Eigenvector Eigenvector • Ex 1: (Verifying eigenvalues and eigenvectors)

32. Note: If has nonzero solutions iff . • Question: Given an nn matrix A, how can you find the eigenvalues and corresponding eigenvectors? • Note: (homogeneous system) • Characteristic equation of AMnn:

33. Sol: Characteristic equation: Eigenvalue: • Ex: (Finding eigenvalues and eigenvectors)

35. Try the solution and plug this into the differential equations: Application Example of Eigenvalue-Eigenvector Problem The equations of motion for identical mass and spring constant can be described by We can obtain Rearrange these to put them into a neater form

36. A nontrivial solution occurs when the determinant is zero , which yields the following solutions (eigenvalues): With the given eigenvalues, we can find the corresponding eigenvectors (normal modes) to be

37. 2.3 Applications of Determinants • Matrix of cofactors of A: • Adjoint matrix of A:

38. Thm 2B.10: The inverse of a matrix given by its adjoint If A is an n × n invertible matrix, then • Ex:

39. Ex: (a) Find the adjoint of A. (b) Use the adjoint of A to find Sol:

40. adjoint matrix of A • Check: cofactor matrix of A inverse matrix of A

41. Thm 2B.11: Cramer’s Rule (this system has a unique solution)

42. ( i.e., )

43. Pf: A x = b,

45. Ex: Use Cramer’s rule to solve the system of linear equations. Sol:

46. Keywords in This Section: • matrix of cofactors : 餘因子矩陣 • adjoint matrix : 伴隨矩陣 • Cramer’s rule : Cramer 法則

47. 2B.4 Geometry of Determinants: Determinants as Size Functions • We have so far only considered whether or not a determinant is zero, here we shall give a meaning to the value of that determinant. O One way to compute the area that it encloses is to draw this rectangle and subtract the area of each subregion.

48. The region formed by and is bigger, by a factor of k, than the shaded region enclosed by and . That is, size ( , ) = k · size( , ). • The properties in the definition of determinants make reasonable postulates for a function that measures the size of the region enclosed by the vectors in the matrix. See this case:

49. Another property of determinants is that they are unaffected by pivoting. Here are before-pivoting and after-pivoting boxes (the scalar used is • k = 0.35). Although the region on the right, the box formed by and , is more slanted than the shaded region, the two have the same base and the same height and hence the same area. This illustrates that

50. That is, we’ve got an intuitive justification to interpret det ( , . . . , ) as the size of the box formed by the vectors. Example The volume of this parallelepiped, which can be found by the usual formula from high school geometry, is 12.