Chapter 3 Determinants 3.1 The Determinant of a Matrix 3.2 Evaluation of a Determinant using Elementary Operations 3.3 Properties of Determinants 3.4 Introduction to Eigenvalues 3.5 Application of Determinants Elementary Linear Algebra R. Larsen et al. (6 Edition) 投影片設計編製者 淡江大學 電機系 翁慶昌 教授 3.1 The Determinant of a Matrix the determinant of a 2 × 2 matrix: a11 a12 A a a 21 22 det(A) | A | a11a22 a21a12 Note: a 11 a 12 a11 a12 a a 21 a 22 21 a22 Elementary Linear Algebra: Section 3.1, p.123 2/62 Ex. 1: (The determinant of a matrix of order 2) 2 3 1 2 2(2) 1(3) 4 3 7 2 1 4 2 2(2) 4(1) 4 4 0 0 3 2 4 0(4) 2(3) 0 6 6 Note: The determinant of a matrix can be positive, zero, or negative. Elementary Linear Algebra: Section 3.1, p.123 3/62 Minor of the entry aij : The determinant of the matrix determined by deleting the ith row and jth column of A a11 a12 a( i 1)1 M ij a( i 1)1 a n1 a1( j 1) a( i 1)( j 1) a( i 1)( j 1) a1( j 1) a1n a( i 1)( j 1) a( i 1) n a( i 1)( j 1) a( i 1) n an ( j 1) an ( j 1) ann Cofactor of aij : Cij ( 1)i j M ij Elementary Linear Algebra: Section 3.1, p.124 4/62 Ex: a11 a12 A a21 a22 a31 a32 a13 a23 a33 a12 M 21 a32 a13 a33 C21 ( 1) 21 M 21 M 21 Elementary Linear Algebra: Section 3.1, p.124 M 22 a11 a13 a31 a33 C22 ( 1) 22 M 22 M 22 5/62 Notes: Sign pattern for cofactors 3 × 3 matrix 4 × 4 matrix n ×n matrix Notes: Odd positions (where i+j is odd) have negative signs, and even positions (where i+j is even) have positive signs. Elementary Linear Algebra: Section 3.1, p.124 6/62 Ex 2: Find all the minors and cofactors of A. 0 2 1 A 3 1 2 4 0 1 Sol: (1) All the minors of A. 1 2 3 2 3 1 M 11 1, M 12 4 5, M 13 0 1 4 0 4 1 M 21 2 1 0 1 2, 2 1 M 31 5, 1 2 Elementary Linear Algebra: Section 3.1, p.125 0 2 0 1 8 M 22 4, M 23 4 1 4 1 0 2 0 1 6 M 32 3, M 33 3 1 3 2 7/62 Sol: (2) All the cofactors of A. Cij (1)i j M ij C11 3 1 4 5, C13 1, C12 4 0 4 1 1 1 2 0 3 2 0 1 0 2 4, C23 C21 2, C22 8 4 1 0 1 4 1 2 1 0 1 2 1 3, C31 5, C 32 3 2 1 2 Elementary Linear Algebra: Section 3.1, p.125 C33 0 2 3 1 6 8/62 Thm 3.1: (Expansion by cofactors) Let A is a square matrix of order n. Then the determinant of A is given by n (a) det(A) | A | aijCij ai1Ci1 ai 2Ci 2 ainCin j 1 (Cofactor expansion along the i-th row, i=1, 2,…, n ) or n (b) det( A) | A | aij Cij a1 j C1 j a2 j C2 j anj Cnj i 1 (Cofactor expansion along the j-th row, j=1, 2,…, n ) Elementary Linear Algebra: Section 3.1, p.126 9/62 Ex: The determinant of a matrix of order 3 a11 a12 a13 A a21 a22 a23 a31 a32 a33 det( A) a11C11 a12C12 a13C13 a21C21 a22C22 a23C23 a31C31 a32C32 a33C33 a11C11 a21C21 a31C31 a12C12 a22C22 a32C32 a13C13 a23C23 a33C33 Elementary Linear Algebra: Section 3.1, Addition 10/62 Ex 3: The determinant of a matrix of order 3 0 2 1 A 3 1 2 4 0 1 Ex 2 C11 1, C12 5, C13 4 C21 2, C22 4, C23 8 C31 5, C 32 3, C33 6 Sol: det(A) a11C11 a12 C12 a13 C13 (0)(1) (2)(5) (1)(4) 14 a21C21 a22 C22 a23C23 (3)(2) (1)(4) (2)(8) 14 a31C31 a32 C32 a33 C33 (4)(5) (0)(3) (1)(6) 14 a11C11 a21C21 a31C31 (0)(1) (3)(2) (4)(5) 14 a12 C12 a22 C22 a32 C32 (2)(5) (1)(4) (0)(3) 14 a13C13 a23 C23 a33C33 (1)(4) (2)(8) (1)(6) 14 Elementary Linear Algebra: Section 3.1, p.126 11/62 Ex 5: (The determinant of a matrix of order 3) 0 2 1 A 3 1 2 det( A) ? 4 4 1 Sol: 2 11 1 2 1 2 3 C11 (1) 1 C12 (1) (1)(5) 5 0 1 4 1 1 3 C13 (1) 3 1 4 4 0 det(A) a11C11 a12C12 a13C13 (0)(1) (2)(5) (1)(4) 14 Elementary Linear Algebra: Section 3.1, p.128 12/62 Notes: The row (or column) containing the most zeros is the best choice for expansion by cofactors . Ex 4: (The determinant of a matrix of order 4) 1 2 1 1 A 0 2 4 3 3 0 0 2 det( A) ? 0 3 0 2 Elementary Linear Algebra: Section 3.1, p.127 13/62 Sol: det(A) (3)(C13 ) (0)(C23 ) (0)(C33 ) (0)(C43 ) 3C13 1 1 3(1)13 0 3 2 2 3 4 2 2 2 2 1 1 22 1 23 1 1 3(0)( 1) (2)( 1) (3)( 1) 4 2 3 2 3 4 30 (2)(1)( 4) (3)( 1)( 7) (3)(13) 39 Elementary Linear Algebra: Section 3.1, p.127 14/62 The determinant of a matrix of order 3: Subtract these three products. a11 a12 a13 A a 21 a 22 a 23 a 31 a 32 a 33 a11 a12 a13 a11 a21 a22 a23 a21 a22 a31 a33 a31 a32 a12 a32 Add these three products. det( A) | A | a11a22 a33 a12 a23a31 a13a21a32 a31a22 a13 a32 a 23 a11 a33a21a12 Elementary Linear Algebra: Section 3.1, p.127 15/62 Ex 5: –4 0 6 0 2 1 0 2 A 3 1 2 3 1 4 4 1 4 4 0 16 –12 det(A) | A | 0 16 12 (4) 0 6 2 Elementary Linear Algebra: Section 3.1, p.128 16/62 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. Note: A matrix that is both upper and lower triangular is called diagonal. Elementary Linear Algebra: Section 3.1, p.128 17/62 Ex: a11 a12 a13 0 a22 a23 0 0 a 33 a11 0 0 a21 a22 0 a a a 31 32 33 a11 0 0 0 a22 0 0 0 a 33 upper triangular lower triangular diagonal Elementary Linear Algebra: Section 3.1, p.128 18/62 Thm 3.2: (Determinant of a Triangular Matrix) If A is an n × n triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is det( A) | A | a11a22 a33 ann Elementary Linear Algebra: Section 3.1, p.129 19/62 Ex 6: Find the determinants of the following triangular matrices. 0 2 4 2 (a) A 5 6 1 5 0 0 0 0 1 0 3 3 Sol: 1 0 (b) B 0 0 0 0 0 0 3 0 0 0 2 0 0 0 4 0 0 0 0 0 0 0 2 (a) |A| = (2)(–2)(1)(3) = –12 (b) |B| = (–1)(3)(2)(4)(–2) = 48 Elementary Linear Algebra: Section 3.1, p.129 20/62 Keywords in Section 3.1: 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 21/62 3.2 Evaluation of a determinant using elementary operations Thm 3.3: (Elementary row operations and determinants) Let A and B be square matrices. ( a ) B rij ( A) det( B) det( A) (i.e. rij ( A) A ) (b) B ri( k ) ( A) det(B) k det(A) (i.e. ri( k ) ( A) k A ) (c) B rij( k ) ( A) det(B) det(A) (i.e. rij( k ) ( A) A ) Elementary Linear Algebra: Section 3.2, p.134 22/62 Ex: 1 A 0 1 4 A1 0 1 2 3 det(A) 2 1 4 2 1 8 12 2 3 0 1 4 1 1 4 A2 1 2 3 A3 2 3 2 1 1 2 1 2 1 2 1 A1 r1( 4) ( A) det(A1) det(r1( 4) ( A)) 4 det(A) (4)(2) 8 A2 r12 ( A) det(A2) det(r12 ( A)) det(A) (2) 2 A3 r12( 2) ( A) det(A3) det(r12( 2) ( A)) det(A) 2 Elementary Linear Algebra: Section 3.2, Addition 23/62 Notes: det(rij ( A)) det(A) det(A) det(rij ( A)) 1 det( ri ( A)) k det( A) det( A) det( ri( k ) ( A)) k (k ) det(rij(k ) ( A)) det(A) det(A) det(rij(k ) ( A)) Elementary Linear Algebra: Section 3.2, pp.134-135 24/62 Note: A row-echelon form of a square matrix is always upper triangular. Ex 2: (Evaluation a determinant using elementary row operations) 2 3 10 A 1 2 2 det( A) ? 0 1 3 Sol: 2 3 10 det(A) 1 0 2 1 r12 1 2 2 2 2 3 10 3 0 1 3 Elementary Linear Algebra: Section 3.2, pp.134-135 25/62 r12( 2 ) 1 2 2 ( 1 ) r2 7 0 7 14 (1)( 0 1 3 ( 1) r23 1 1 7 1 2 2 ) 0 1 2 0 1 3 1 2 2 7 0 1 2 (7)(1)(1)(1) 7 0 0 1 Elementary Linear Algebra: Section 3.2, pp.134-135 26/62 Notes: EA E A (1) E Rij E Rij 1 EA rij A A Rij A E A (k ) (2) E Ri( k ) E Ri k EA ri( k ) A k A Ri( k ) A E A (k ) (3) E Rij( k ) E Rij 1 EA rij( k ) A 1 A Rij( k ) A E A Elementary Linear Algebra: Section 3.2, Addition 27/62 Determinants and elementary column operations Thm: (Elementary column operations and determinants) Let A and B be square matrices. (a) B cij ( A) det( B) det( A) (i.e. cij ( A) A ) (b) B ci( k ) ( A) det(B) k det(A) (i.e. ci( k ) ( A) k A ) (c) B cij( k ) ( A) det(B) det(A) (i.e. cij( k ) ( A) A ) Elementary Linear Algebra: Section 3.2, p.135 28/62 Ex: 2 1 3 A 4 0 1 0 0 2 det(A) 8 1 1 3 A1 2 0 1 0 0 2 1 2 3 A2 0 4 1 0 0 2 2 1 0 A3 4 0 1 0 0 2 1 1 A1 c ( A) det( A1) det( c1 ( A)) det( A) ( )( 8) 4 2 2 A2 c12 ( A) det(A2) det(c12 ( A)) det(A) (8) 8 1 ( ) 2 1 ( 4) A3 c23( 3) ( A) det(A3) det(c23( 3) ( A)) det(A) 8 Elementary Linear Algebra: Section 3.2, p.135 29/62 Thm 3.4: (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). Elementary Linear Algebra: Section 3.2, p.136 30/62 Ex: 1 2 3 1 4 0 1 1 1 0 0 0 0 4 5 6 2 5 0 0 3 6 0 2 2 2 0 4 5 6 1 4 2 1 5 2 0 1 6 2 1 2 3 4 5 6 0 2 4 6 Elementary Linear Algebra: Section 3.2, Addition 1 8 4 2 10 5 0 3 12 6 31/62 Note: Cofactor Expansion Row Reduction Order n Additions Multiplications Additions Multiplications 3 5 9 5 10 5 119 205 30 45 10 3,628,799 6,235,300 285 339 Elementary Linear Algebra: Section 3.2, p.138 32/62 Ex 5: (Evaluating a determinant) 2 3 5 A 2 4 1 6 3 0 3 5 2 Sol: 3 5 4 det(A) 2 4 1 2 4 3 3 0 6 3 0 0 ( 2) C13 5 4 (3)(1) (3)(1) 3 4 3 3 5 2 ( 54 ) 3 5 2 r12 det(A) 2 4 1 52 0 53 3 0 6 3 0 6 31 2 5 3 5 3 (5)(1) (5)( ) 3 3 6 5 1 2 Elementary Linear Algebra: Section 3.2, p.138 33/62 Ex 6: (Evaluating a determinant) 2 0 1 2 1 3 A 1 0 1 3 1 2 1 1 3 3 2 2 1 2 3 4 3 2 0 Sol: 2 0 1 2 1 3 det(A) 1 0 1 3 1 2 1 1 3 2 (1)(1)2 2 1 1 3 3 2 2 2 1 2 2 3 (1) 1 4 3 rr24( 1) 1 2 0 25 3 1 1 5 0 3 2 6 0 Elementary Linear Algebra: Section 3.2, p.139 0 1 1 3 0 1 0 5 0 0 3 2 2 1 2 3 6 4 0 1 2 3 4 1 34/62 8 1 ( 3 ) C41 8 1 13 5 0 0 1 3 5(1) 3 2 8 1 3 0 0 5 2 3 (1)(1) 44 8 1 2 (1) 8 1 2 r21 6 4 13 5 6 13 5 6 0 1 8 1 13 5 (5)( 27) 135 Elementary Linear Algebra: Section 3.2, p.140 35/62 3.3 Properties of Determinants Thm 3.5: (Determinant of a matrix product) det (AB) = det (A) det (B) Notes: (1) det (EA) = det (E) det (A) (2) det( A B) det( A) det( B) (3) a11 a12 a22 b22 a31 a22 b22 a32 a13 a11 a12 a13 a11 a12 a13 a23 b23 a21 a22 a23 b21 b22 b23 a31 a32 a33 a31 a32 a33 a33 Elementary Linear Algebra: Section 3.3, p.143 36/62 Ex 1: (The determinant of a matrix product) 1 2 2 A 0 3 2 1 0 1 1 2 0 B 0 1 2 3 1 2 Find |A|, |B|, and |AB| Sol: 1 2 2 | A | 0 1 3 0 2 7 1 Elementary Linear Algebra: Section 3.3, p.143 2 0 1 | B | 0 1 2 11 3 1 2 37/62 1 8 4 1 1 2 2 2 0 AB 0 3 2 0 1 2 6 1 10 1 0 1 3 1 2 5 1 1 8 4 1 | AB | 6 1 10 77 5 1 1 Check: |AB| = |A| |B| Elementary Linear Algebra: Section 3.3, p.143 38/62 Thm 3.6: (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) Ex 2: 10 20 40 A 30 0 50, 20 30 10 Find |A|. 1 2 4 3 0 5 5 2 3 1 Sol: 1 2 4 1 2 4 3 A 10 3 0 5 (1000)(5) 5000 A 10 3 0 5 2 3 1 2 3 1 Elementary Linear Algebra: Section 3.3, p.144 39/62 Thm 3.7: (Determinant of an invertible matrix) A square matrix A is invertible (nonsingular) if and only if det (A) 0 Ex 3: (Classifying square matrices as singular or nonsingular) 0 2 1 A 3 2 1 3 2 1 0 2 1 B 3 2 1 3 2 1 Sol: A 0 A has no inverse (it is singular). B 12 0 B has an inverse (it is nonsingular). Elementary Linear Algebra: Section 3.3, p.145 40/62 Thm 3.8: (Determinant of an inverse matrix) 1 If A is invertible, then det( A 1 ) . det( A) Thm 3.9: (Determinant of a transpose) If A is a square matrix, then det( AT ) det( A). Ex 4: Sol: 1 0 3 A 0 1 2 2 1 0 1 0 3 | A | 0 1 2 4 2 1 0 Elementary Linear Algebra: Section 3.3, pp.146-148 1 (a) A ? (b) AT ? 1 1 A 4 AT A 4 A1 41/62 Equivalent conditions for a nonsingular matrix: If A is an n × n matrix, then the following statements are equivalent. (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. (4) A is row-equivalent to In (5) A can be written as the product of elementary matrices. (6) det (A) 0 Elementary Linear Algebra: Section 3.3, p.147 42/62 Ex 5: Which of the following system has a unique solution? x3 x3 x3 1 4 4 2 x2 x3 1 3x1 2 x2 x3 4 3x1 2 x2 x3 4 (a) 3x1 3x1 (b) 2 x2 2 x2 2 x2 Elementary Linear Algebra: Section 3.3, p.148 43/62 Sol: (a) Ax b A 0 This system does not have a unique solution. (b) Bx b B 12 0 This system has a unique solution. Elementary Linear Algebra: Section 3.3, p.148 44/62 3.4 Introduction to Eigenvalues Eigenvalue problem: If A is an nn matrix, do there exist n1 nonzero matrices x such that Ax is a scalar multiple of x? Eigenvalue and eigenvector: A:an nn matrix :a scalar x: a n1 nonzero column matrix Eigenvalue Ax x (The fundamental equation for the eigenvalue problem) Eigenvector Elementary Linear Algebra: Section 3.4, p.152 45/62 Ex 1: (Verifying eigenvalues and eigenvectors) 1 4 A 2 3 1 x1 1 2 x2 1 Eigenvalue 1 4 1 5 1 Ax1 5 5 x1 2 3 1 5 1 Eigenvector Eigenvalue 1 4 2 2 2 Ax 2 1 (1) x2 2 3 1 1 1 Eigenvector Elementary Linear Algebra: Section 3.4, p.152 46/62 Question: Given an nn matrix A, how can you find the eigenvalues and corresponding eigenvectors? Note: Ax x (I A) x 0 (homogeneous system) If (I A) x 0 has nonzero solutions iff det(I A) 0. Characteristic equation of AMnn: det( I A) (I A) n cn1n1 c1 c0 0 Elementary Linear Algebra: section 3.4, p.153 47/62 Ex 2: (Finding eigenvalues and eigenvectors) 1 4 A 2 3 Sol: Characteristic equation: 1 4 I A 2 3 2 4 5 ( 5)( 1) 0 5, 1 Eigenvalues: 1 5, 2 1 Elementary Linear Algebra: Section 3.4, p.153 48/62 (1)1 5 (2)2 1 4 4 x1 0 (1I A) x 2 2 x2 0 x1 t 1 x t t 1, t 0 2 2 4 x1 0 (2 I A) x 2 4 x2 0 x1 2t 2 x t t 1, t 0 2 Elementary Linear Algebra: Section 3.4, p.153 49/62 Ex 3: (Finding eigenvalues and eigenvectors) 1 2 2 A 1 2 1 1 1 0 Sol: Characteristic equation: 1 I A 1 1 2 2 2 1 ( 1)( 1)( 3) 0 1 T heeigenvalues : 1 1, 2 1, 3 3 Elementary Linear Algebra: Section 3.4, p.154 50/62 0 2 2 1 0 2 1I A 1 1 1 ~ 0 1 1 1 1 0 0 0 1 1 1 x1 2t 2 x t eigenvect ors : t 1 , t 0 2 x3 t 1 2 2 2 1 0 2 2 I A 1 3 1 ~ 0 1 1 1 1 0 0 0 1 x1 2t 2 x t eigenvectors : t 1, t 0 2 x3 t 1 2 1 Elementary Linear Algebra: Section 3.4, p.155 51/62 3 3 2 2 2 1 0 2 3I A 1 1 1 ~ 0 1 1 1 3 0 0 0 1 x1 2t x t eigenvectors : 2 x3 t Elementary Linear Algebra: Section 3.4, p.156 2 t 1 , t 0 1 52/62 3.5 Applications of Determinants Matrix of cofactors of A: C11 C12 C1n C C C 2n Cij 21 22 Cn1 Cn 2 Cnn Cij (1)i j Mij Adjoint matrix of A: adj( A) Cij T C11 C 12 C1n Elementary Linear Algebra: Section 3.5, p.158 C21 Cn1 C22 Cn 2 C2 n Cnn 53/62 Thm 3.10: (The inverse of a matrix given by its adjoint) If A is an n × n invertible matrix, then 1 A adj( A) det(A) 1 Ex: a b A c d det( A) ad bc d b adj( A) c a Elementary Linear Algebra: Section 3.5, p.159 1 A adj( A) det A 1 d b ad bc c a 1 54/62 Ex 1 & Ex 2: 2 1 3 A 0 2 1 1 0 2 (a) Find the adjoint of A. (b) Use the adjoint of A to find A1 i j C ( 1 ) M ij Sol: ij 2 1 0 1 0 2 C11 4, C12 1, C13 2 0 2 1 2 1 0 C21 C31 3 2 0 2 3 6, 2 2 1 7, Elementary Linear Algebra: Section 3.5, p.160 C22 C 32 1 2 1 2 0, C23 1 3 1 0 3 1 3 2 1, C33 1 0 2 1 2 0 55/62 cofactor matrix of A adjoint matrix of A 4 1 2 Cij 6 0 3 7 1 2 adj( A) Cij inverse matrix of A 1 A1 adj( A) det A 4 6 7 43 2 13 1 0 1 13 0 2 3 2 23 1 Check: T 4 6 7 1 0 1 2 3 2 det A 3 1 3 2 3 7 3 AA 1 I Elementary Linear Algebra: Section 3.5, p.160 56/62 Thm 3.11: (Cramer’s Rule) a11 x1 a12 x2 a1n xn b1 a21 x1 a22 x2 a2 n xn b2 an1 x1 an 2 x2 ann xn bn A aij nn A(1) , A( 2) ,, A( n ) Ax b det(A) a1n a11 a12 a21 a22 a2 n an1 an 2 b1 x1 b x 2 x 2 b b x n n 0 (this system has a unique solution) ann Elementary Linear Algebra: Section 3.5, p.163 57/62 Aj A(1) , A( 2) ,, A( j 1) , b, A( j 1) ,, A(n) a11 a1( j 1) b1 a1( j 1) a1n a a b a a 21 2 ( j 1) 2 2 ( j 1) 2n a a b a a n ( j 1) n n ( j 1) nn n1 ( i.e. det( Aj ) b1C1 j b2C2 j bnCnj ) det( A j ) xj , det( A) j 1, 2 ,, n Elementary Linear Algebra: Section 3.5, p.163 58/62 Pf: det(A) 0 A x = b, 1 x A b adj( A)b det( A) C11 C21 Cn1 b1 1 C12 C22 Cn 2 b2 det (A) C C C 2n nn bn 1n 1 b1C11 1 b1C12 det( A) b C 1 1n b2C21 b2C22 b2C2 n Elementary Linear Algebra: Section 3.5, p.163 bn Cn1 bnCn 2 bn Cnn 59/62 xj 1 (b1C1 j b2C2 j bnCnj ) det(A) det(A j ) det(A) j 1,2 ,, n Elementary Linear Algebra: Section 3.5, p.163 60/62 Ex 4: Use Cramer’s rule to solve the system of linear equations. x 2 y 3z 1 2x z 0 3x 4 y 4 z 2 Sol: 1 det (A) 2 3 2 0 4 3 1 10 4 1 det( A1 ) 0 0 2 4 1 1 1 3 det( A2 ) 2 3 0 2 det( A1 ) 4 x det( A) 5 1 15, 4 det( A3 ) 2 3 det( A2 ) 3 y det( A) 2 Elementary Linear Algebra: Section 3.5, p.163 3 2 2 1 8 4 1 0 0 16 4 2 det( A3 ) 8 z det( A) 5 61/62 Keywords in Section 3.5: matrix of cofactors : 餘因子矩陣 adjoint matrix : 伴隨矩陣 Cramer’s rule : Cramer 法則 62/62