1 資訊科學數學 14 : Determinants & Inverses 陳光琦助理教授 (Kuang-Chi Chen)

2 Linear Equations and Matrices Determinants

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. 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 Case 1: 1  1 matrices - If A = (a), then A will have a multiplicative inverse iff a≠0. - If A = (a), then A will have a multiplicative inverse iff a≠0. - A is nonsingular iff det(A)≠0. - A is nonsingular iff det(A)≠0.

4 2  2 Matrices Case 2: 2  2 matrices Case 2: 2  2 matrices - Let A =. - Let A =. - A will be nonsingular iff det(A) = a 11 a 22 – a 12 a 21 ≠ 0. - A will be nonsingular iff det(A) = a 11 a 22 – a 12 a 21 ≠ 0.

5 3  3 Matrices Case 3: 3  3 matrices Case 3: 3  3 matrices - Let A =. - Let A =. - A will be nonsingular iff - A will be nonsingular iff det(A) = a 11 a 22 a 33 + a 12 a 31 a 23 + a 13 a 21 a 32 – a 11 a 32 a 23 – a 12 a 21 a 33 – a 13 a 31 a 22 ≠ 0. det(A) = a 11 a 22 a 33 + a 12 a 31 a 23 + a 13 a 21 a 32 – a 11 a 32 a 23 – a 12 a 21 a 33 – a 13 a 31 a 22 ≠ 0.

6 Example 4 & 5 Example 4 Example 4 If A = [a 11 ] is a 1  1 matrix, then det(A) = a 11. If A = [a 11 ] is a 1  1 matrix, then det(A) = a 11. Example 5 Example 5 If If ⇒ det(A) = a 11 a 22 – a 12 a 21 ⇒ det(A) = a 11 a 22 – a 12 a 21 ⇒ det(A) = (2)(5) – (-3)(4) = 22 ⇒ det(A) = (2)(5) – (-3)(4) = 22

7 Example 6 & 7 Example 6 Example 6If ⇒ det(A) = a 11 a 22 a 33 + a 12 a 31 a 23 + a 13 a 21 a 32 ⇒ det(A) = a 11 a 22 a 33 + a 12 a 31 a 23 + a 13 a 21 a 32 – a 11 a 32 a 23 – a 12 a 21 a 33 – a 13 a 31 a 22 – a 11 a 32 a 23 – a 12 a 21 a 33 – a 13 a 31 a 22 Example 7 Example 7If ⇒ det(A) = (1)(1)(2) + (3)(2)(1) + (2)(3)(3) ⇒ det(A) = (1)(1)(2) + (3)(2)(1) + (2)(3)(3) – (3)(1)(3) – (1)(1)(3) – (2)(2)(2) = 6 – (3)(1)(3) – (1)(1)(3) – (2)(2)(2) = 6

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

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

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

11 Example 9 & 10 Example 9 Example 9If Example 10 Example 10If

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

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

14 Example 13 Example 13 Example 13

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

16 Theorem 3.7 Theorem 3.7 Theorem 3.7 If a matrix A = [a ij ] is upper (lower) triangular, then, then det(A) = a 11 a 22 … a nn. If a matrix A = [a ij ] is upper (lower) triangular, then, then det(A) = a 11 a 22 … a nn. Corollary 1.3 Corollary 1.3 The determinant of a diagonal matrix is the product of the entries on its main diagonal. The determinant of a diagonal matrix is the product of the entries on its main diagonal.

17 Example 15 Example 15 Example 15

18 Elementary Operations Elementary row and elementary column operations Elementary row and elementary column operations I - Interchange rows (columns) i and j : I - Interchange rows (columns) i and j : r i ⇔ r j (c i ⇔ c j ) r i ⇔ r j (c i ⇔ c j ) II - Replace row (column) i by a nonzero value k times row (column) i : II - Replace row (column) i by a nonzero value k times row (column) i : kr i ⇔ r i (kc i ⇔ c i ) kr i ⇔ r i (kc i ⇔ c i ) III - Replace row (column) j by a nonzero value k times row (column) i+ row (column) j : III - Replace row (column) j by a nonzero value k times row (column) i+ row (column) j : kr i + r j ⇔ r j (kc i + c j ⇔ c j ) kr i + r j ⇔ r j (kc i + c j ⇔ c j )

19 … then …

20 Example 16 E.g. 16 E.g. 16

21 Example 16 (cont’d)

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

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

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

25 Example 18 Example 18 Example 18

26 Cofactor Expression and Applications Cofactor Expression and Applications

Cofactor Expression and Applications Cofactor expression and applications Definition – Minor and cofactor Definition – Minor and cofactor Let A = [a ij ] be an n  n matrix. Let M ij be the (n-1)  (n-1) submatrix of A obtained by deleting the i th row and j th column of A. The determinant det(M ij ) is called the minor of a ij. The cofactor A ij of a ij is defined as Let A = [a ij ] be an n  n matrix. Let M ij be the (n-1)  (n-1) submatrix of A obtained by deleting the i th row and j th column of A. The determinant det(M ij ) is called the minor of a ij. The cofactor A ij of a ij is defined as

28 Example 1 E.g. 1 E.g. 1 Let Let

29 Theorem 3.9 Theorem 3.9 Theorem 3.9 Let A = [a ij ] be an n  n matrix. Then for each 1≤ i ≤ n, for each 1≤ i ≤ n, det(A) = a i1 A i1 + a i2 A i2 + … + a in A in, and det(A) = a i1 A i1 + a i2 A i2 + … + a in A in, and for each 1≤ j ≤ n, for each 1≤ j ≤ n, det(A) = a 1j A 1j + a 2j A 2j + … + a nj A nj. det(A) = a 1j A 1j + a 2j A 2j + … + a nj A nj.

30 Example 2 To evaluate the determinant

31 Example 3 Consider the determinant of the matrix

32 Theorem 3.10 Theorem 3.10 Theorem 3.10 If A = [a ij ] be an n  n matrix, then If A = [a ij ] be an n  n matrix, then a i1 A k1 + a i2 A k2 + … + a in A kn = 0, for i≠k, a i1 A k1 + a i2 A k2 + … + a in A kn = 0, for i≠k, a 1j A 1k + a 2j A 2k + … + a nj A nk = 0, for j≠k. a 1j A 1k + a 2j A 2k + … + a nj A nk = 0, for j≠k.

33 Example 4 E.g. 4 E.g. 4

34 Adjoint Definition – Adjoint Definition – Adjoint Let A = [a ij ] be an n  n matrix. The n  n matrix adj A, called the adjoint of A, is the matrix whose j, i th element is the cofactor A ij of a ij. Thus Let A = [a ij ] be an n  n matrix. The n  n matrix adj A, called the adjoint of A, is the matrix whose j, i th element is the cofactor A ij of a ij. Thus

35 Remark Remark Remark The adjoint of A is formed by taking the transpose of the matrix of cofactors A ij of the elements of A. The adjoint of A is formed by taking the transpose of the matrix of cofactors A ij of the elements of A.

36 Example 5 Example 5 Example 5 Compute adj A Compute adj A

37 Solution

38 Theorem 3.11 Theorem 3.11 Theorem 3.11 If A = [a ij ] be an n  n matrix, then If A = [a ij ] be an n  n matrix, then A(adj A) = (adj A)A = det(A) I n. A(adj A) = (adj A)A = det(A) I n.

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

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

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

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

43 Example 8 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 = [b 1, b 2, b 3, b 4 ] 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 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 = I n (c) A solution to the given system is given by x = A -1 b (d) No. The solution is unique. (e) Yes.

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

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

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

48 Cramer ’ s Rule Theorem 3.13 (Cramer’s Rule) Let a 11 x 1 + a 12 x 2 + … + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + … + a 2n x n = b 2 a 21 x 1 + a 22 x 2 + … + a 2n x n = b 2 … a n1 x 1 + a n2 x 2 + … + a nn x n = b n a n1 x 1 + a n2 x 2 + … + a nn x n = b nThen, x 1 = det(A 1 )/det(A), x 2 = det(A 2 )/det(A), …, x 1 = det(A 1 )/det(A), x 2 = det(A 2 )/det(A), …, x n = det(A n )/det(A). x n = det(A n )/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, x i = det(A i )/det(A), x i = det(A i )/det(A), where A i is the matrix obtained from A by replacing the i th column of A by b. where A i is the matrix obtained from A by replacing the i th column of A by b.

50 Example 9 Consider the following linear system: Consider the following linear system: -2x 1 + 3x 2 – x 3 = 1 -2x 1 + 3x 2 – x 3 = 1 x 1 + 2x 2 – x 3 = 4 x 1 + 2x 2 – x 3 = 4 -2x 1 – 2x 2 + x 3 = -3 -2x 1 – 2x 2 + x 3 = -3 Then Then

51 Example 9 (cont ’ d) Hence,

52 Polynomial Interpolation Revisited Polynomial Interpolation Revisited Polynomial Interpolation Revisited To find a quadratic polynomial that interpolates the following points: To find a quadratic polynomial that interpolates the following points: (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ), (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ), where x 1 ≠x 2, x 1 ≠ x 3, x 2 ≠ x 3. where x 1 ≠x 2, x 1 ≠ x 3, x 2 ≠ x 3.

53 … more … The polynomial has the form: y = a 2 x 2 + a 1 x + a 0. y = a 2 x 2 + a 1 x + a 0. The corresponding linear system y 1 = a 2 x a 1 x 1 + a 0, y 1 = a 2 x a 1 x 1 + a 0, y 2 = a 2 x a 1 x 2 + a 0, y 2 = a 2 x a 1 x 2 + a 0, y 3 = a 2 x a 1 x 3 + a 0. y 3 = a 2 x a 1 x 3 + a 0.

54 … more … The coefficient matrix The Vandermount determinant (x 1 – x 2 )( x 1 – x 3 )( x 2 – x 3 ) (x 1 – x 2 )( x 1 – x 3 )( x 2 – x 3 )

55

56 Linear Equations and Matrices LU-Factorization

57 LU-Factorization 1.8 LU-Factorization 1.8 LU-Factorization If a square matrix can be reduced to upper triangular form using only 3 row operations, then it is possible to represent the reduction process in terms of a matrix factorization. If a square matrix can be reduced to upper triangular form using only 3 row operations, then it is possible to represent the reduction process in terms of a matrix factorization.

58 Type I Operation An elementary matrix of type I is a matrix obtained by interchanging two rows of identity matrix I. Example

59 Type II Operation An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant. Example

60 Type III Operation An elementary matrix of type III is a matrix obtained from I by adding a multiple of one row to another row. Example

61 In General In general, the elementary matrix by adding m times of row i to row j Row j Column i a ji

62 Theorem Theorem Theorem If A and B are nonsingular square matrices, then AB is also nonsingular. If A and B are nonsingular square matrices, then AB is also nonsingular. i.e. (AB) -1 = B -1 A -1. i.e. (AB) -1 = B -1 A -1. In general, if E 1, E 2, …, E k are all nonsingular, then the product E 1 E 2 … E k is also nonsingular and In general, if E 1, E 2, …, E k are all nonsingular, then the product E 1 E 2 … E k is also nonsingular and (E 1 E 2 … E k ) -1 = E k -1 … E 2 -1 E (E 1 E 2 … E k ) -1 = E k -1 … E 2 -1 E 1 -1.

63 Row Equivalence A matrix B is row equivalent to A if there exists a finite sequence E 1 E 2 … E k of elementary matrices such that B = E k … E 2 E 1 A. A matrix B is row equivalent to A if there exists a finite sequence E 1 E 2 … E k of elementary matrices such that B = E k … E 2 E 1 A.

64 LU-Factorization LU-Factorization LU-Factorization If a square matrix can be reduced to upper triangular form using only 3 row operations, then it is possible to represent the reduction process in terms of a matrix factorization. If a square matrix can be reduced to upper triangular form using only 3 row operations, then it is possible to represent the reduction process in terms of a matrix factorization.

65 Example Let Let

66 (cont ’ d)

67 (cont ’ d)