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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

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2 Linear Equations and Matrices Determinants

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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.

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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.

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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.

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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

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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

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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 ).

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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)

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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.

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

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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

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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

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14 Example 13 Example 13 Example 13

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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

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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.

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17 Example 15 Example 15 Example 15

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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 )

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19 … then …

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

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21 Example 16 (cont’d)

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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

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

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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 | )

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25 Example 18 Example 18 Example 18

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26 Cofactor Expression and Applications Cofactor Expression and Applications

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27 3.2 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

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

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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.

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30 Example 2 To evaluate the determinant

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31 Example 3 Consider the determinant of the matrix

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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.

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

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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

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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.

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

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37 Solution

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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.

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

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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

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

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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.

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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?

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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.

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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 = 0. 3. 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.

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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

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

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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).

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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.

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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

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51 Example 9 (cont ’ d) Hence,

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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.

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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 1 2 + a 1 x 1 + a 0, y 1 = a 2 x 1 2 + a 1 x 1 + a 0, y 2 = a 2 x 2 2 + a 1 x 2 + a 0, y 2 = a 2 x 2 2 + a 1 x 2 + a 0, y 3 = a 2 x 3 2 + a 1 x 3 + a 0. y 3 = a 2 x 3 2 + a 1 x 3 + a 0.

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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 )

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56 Linear Equations and Matrices LU-Factorization

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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.

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

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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

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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

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

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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 1 -1. (E 1 E 2 … E k ) -1 = E k -1 … E 2 -1 E 1 -1.

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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.

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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.

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65 Example Let Let

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66 (cont ’ d)

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67 (cont ’ d)

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