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資訊科學數學 14 : Determinants & Inverses

資訊科學數學 14 : Determinants & Inverses. 陳光琦助理教授 (Kuang-Chi Chen) chichen6@mail.tcu.edu.tw. Linear Equations and Matrices Determinants. 3.1 Determinants.

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資訊科學數學 14 : Determinants & Inverses

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  1. 資訊科學數學14 :Determinants & Inverses 陳光琦助理教授 (Kuang-Chi Chen) chichen6@mail.tcu.edu.tw

  2. Linear Equations and MatricesDeterminants

  3. 3.1 Determinants • With each nn matrix A it is possible to associate a scalar det(A), called the determinant of the matrix, whose value will tell us whether the matrix is singular or not. • Case 1: 11 matrices - If A = (a), then A will have a multiplicative inverse iff a≠0 . - A is nonsingular iff det(A)≠0 .

  4. 22 Matrices • Case 2: 22 matrices - Let A = . - A will be nonsingular iff det(A) = a11a22 – a12a21≠ 0 .

  5. 33 Matrices • Case 3: 33 matrices - Let A = . - A will be nonsingular iff det(A) = a11a22a33 + a12a31a23 + a13a21a32 – a11a32a23 – a12a21a33 – a13a31a22 ≠ 0 .

  6. Example 4 & 5 • Example 4 If A = [a11] is a 11 matrix, then det(A) = a11 . • Example 5 If ⇒ det(A) = a11a22 – a12a21 ⇒det(A) = (2)(5) – (-3)(4) = 22

  7. Example 6 & 7 • Example 6 If ⇒ det(A) =a11a22a33 + a12a31a23 + a13a21a32 – a11a32a23 – a12a21a33 – a13a31a22 • Example 7 If ⇒ det(A) = (1)(1)(2) + (3)(2)(1)+ (2)(3)(3) – (3)(1)(3)– (1)(1)(3)– (2)(2)(2) = 6

  8. Properties of Determinants • Theorem 3.1 The determinants of a matrix and its transpose are equal, i.e., det(A) = det(AT).

  9. Example 8 • Example 8 If ⇒ det(AT) = (1)(1)(2) + (3)(1)(2)+ (2)(3)(3) – (3)(1)(3)– (1)(1)(3)– (2)(2)(2) = 6 = det(A)

  10. Theorem 3.2 & 3.3 • Theorem 3.2 If matrix B results from matrix A by interchanging two rows (or two columns) of A, then det(B) = -det(A). • Theorem 3.3 If two rows (or columns) of A are equal, then det(A) = 0.

  11. Example 9 & 10 • Example 9 If • Example 10 If

  12. Theorem 3.4 • Theorem 3.4 If a row (or column) of A consists entirely of zeros, then det(A) = 0. • Example 11

  13. Theorem 3.5 • Theorem 3.5 If B is obtained from A by multiplying a row (column) of A by a real number c, then det(B) = c det(A) . • Example 12

  14. Example 13 • Example 13

  15. Theorem 3.6 • Theorem 3.6 If B = [bij] is obtained from A = [aij] by adding to each element of the rth row (column) of A a constant c times the corresponding element of the sth row (column) r≠s of A, then det(B) = det(A) . • Example 14

  16. Theorem 3.7 • Theorem 3.7 If a matrix A = [aij] is upper (lower) triangular, then, then det(A) = a11 a22 … ann . • Corollary 1.3 The determinant of a diagonal matrix is the product of the entries on its main diagonal.

  17. Example 15 • Example 15

  18. Elementary Operations • Elementary row and elementary column operations I - Interchange rows (columns) i and j : ri ⇔ rj (ci ⇔ cj ) II - Replace row (column) i by a nonzero value k times row (column) i : kri ⇔ ri (kci ⇔ ci ) III - Replace row (column) j by a nonzero value k times row (column) i+ row (column) j : kri + rj ⇔ rj (kci + cj ⇔ cj )

  19. … then …

  20. Example 16 • E.g. 16

  21. Example 16(cont’d)

  22. Theorem 3.8 • Theorem 3.8 The determinant of a product of two matrices is the product of their determinants det(AB) = det(A)det(B) . • Example 17

  23. Example 17 (cont’d) • Remark AB≠BA |BA| = |B| |A|= -10 = |AB|

  24. Corollary 3.2 • Corollary 3.2 If A is nonsingular, then det(A) ≠ 0, thus det(A-1) = 1/det(A). If A is singular, then det(A) = 0 ( 1 = |I| = |AA-1| = |A| |A-1| )

  25. Example 18 • Example 18

  26. Cofactor Expression and Applications

  27. 3.2 Cofactor Expression and Applications Cofactor expression and applications • Definition – Minor and cofactor Let A = [aij] be an nn matrix. Let Mij be the (n-1) (n-1) submatrix of A obtained by deleting the ith row and jth column of A. The determinant det(Mij) is called the minor of aij. The cofactor Aij of aij is defined as

  28. Example 1 • E.g. 1 Let

  29. Theorem 3.9 • Theorem 3.9 Let A = [aij] be an nn matrix. Then for each 1≤ i ≤ n, det(A) = ai1Ai1 + ai2Ai2 + … + ainAin , and for each 1≤ j ≤ n, det(A) = a1jA1j + a2jA2j + … + anjAnj .

  30. Example 2 To evaluate the determinant

  31. Example 3 Consider the determinant of the matrix

  32. Theorem 3.10 • Theorem 3.10 If A = [aij] be an nn matrix, then ai1Ak1 + ai2Ak2 + … + ainAkn = 0, for i≠k , a1jA1k + a2jA2k + … + anjAnk = 0, for j≠k .

  33. Example 4 • E.g. 4

  34. Adjoint • Definition – Adjoint Let A = [aij] be an nn matrix. The nn matrix adjA, called the adjoint of A, is the matrix whose j, ith element is the cofactor Aij of aij . Thus

  35. Remark • Remark The adjoint of A is formed by taking the transpose of the matrix of cofactorsAij of the elements of A.

  36. Example 5 • Example 5 Compute adj A

  37. Solution

  38. Theorem 3.11 • Theorem 3.11 If A = [aij] be an nn matrix, then A(adj A) = (adj A)A = det(A) In .

  39. Example 6 • E.g. 6 Consider the matrix

  40. Corollary 3.3 • Corollary 3.3 If A = [aij] be an nn matrix and det(A)≠0, then

  41. Example 7 • Example 7 Consider the matrix Then det(A) = -94, and

  42. Theorem 3.12 • Theorem 3.12 A matrix A = [aij] is nonsingular iff det(A) ≠ 0. • Corollary 3.4 For an nn matrix A, the homogeneous system Ax = 0 has a nontrival solution iff det(A) = 0.

  43. Example 8 • Example 8 Let A be a 4x4 matrix with det(A) = -2 (a) describe the set of all solutions to the homogeneous system Ax = 0. (b) If A is transformed to reduced row echelon form B, what is B? (c) Given an expression for a solution to the linear system Ax = b, where b = [b1 , b2 , b3 , b4 ]T . (d) Can the linear system Ax = b have more than one solution? Explain. (e) Does A-1 exist?

  44. Solutions of Example 8 • Solutions (a) Since det(A)≠0, Ax = 0 has only the trivial solution. (b) Since det(A)≠0, A is a nonsingular matrix, so B = In (c) A solution to the given system is given by x = A-1b (d) No. The solution is unique. (e) Yes.

  45. Nonsingular Equivalence • List of nonsingular equivalence The following statements are equivalent. 1.A is nonsingular. 2.x = 0 is the only solution to Ax = 0. 3.A is row equivalence to In . 4. The linear system Ax = b has a unique solution for every n1 matrix b. 5. det(A)≠0 .

  46. Determinants • Linearly independent • Nonsingular • Trivial solution x = 0 to Ax = 0 • det(A) ≠ 0

  47. Determinants • Linearly dependent • Singular • Nontrivial solution to Ax = 0 • det(A) = 0

  48. Cramer’s Rule Theorem 3.13 (Cramer’s Rule) Let a11x1 + a12x2 + … + a1nxn = b1 a21x1 + a22x2 + … + a2nxn = b2 … an1x1 + an2x2 + … + annxn = bn Then, x1 = det(A1)/det(A) , x2 = det(A2)/det(A) , … , xn = det(An)/det(A) .

  49. Cramer’s Rule Cramer’s Rule for solving the linear system Ax = b, where A is nn, is as follows: Step 1. Compute det(A). If det(A) = 0, Cramer’s rule is not applicable. Use Gauss-Jordan Reduction. Step 2. If det(A)≠0, for each i, xi = det(Ai)/det(A) , where Ai is the matrix obtained from A by replacing the ith column of A by b.

  50. Example 9 • Consider the following linear system: -2x1 + 3x2 – x3 = 1 x1 + 2x2 – x3 = 4 -2x1 – 2x2 + x3 = -3 Then

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