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Chapter 2---Section 1 Chapter 2 The Determinants of a Matrix Determinants With each square matrix it is possible to associate a real number called the determinant of the matrix. The value of this number tells us whether the matrix is nonsingular. §1. The Determinant of a Matrix New words and phrases Determinant 行列式 Minor 余子式 Cofactor 代数余子式 1.1 Introduction and Definition First we consider three simple cases. Notice that A is nonsingular if and only if A is row equivalent to I. Case 1 1x1 Matrices If A=(a) , we define det(A)=a. A is nonsingular if and only if det(A) is not zero. Case 2 2x2 Matrices If we use row operations, we see that A is nonsingular if and only if a11a22 a12 a21 0 . We define Chapter 2---Section 1 The Determinants of a Matrix det(A)= a11a22 a12 a21 . We can use the determinants of 1x1 matrices to express det(A) Notation We can refer to the determinant of a specific matrix by enclosing the array between vertical lines. a11 a12 a21 a22 a11a22 a12 a21 Case 3 3x3 Matrices This case is left to students. In this case, if we define that det(A)= a11a22 a33 a12a23a31 a13a21a32 a13a22a31 a12a21a33 a11a23a32 Then we see that A is nonsingular if and only if det(A) is not zero. a11 a12 a13 a21 a22 a23 a31 a32 a33 We can use the determinants of 2x2 matrices to express det(A). ★Definition Let A aij be an nxn matrix and let M ij denote the (n-1)x(n-1) matrix obtained from A by deleting the row and column containing aij . The determinant of M ij is called the minor of aij . We define the cofactor Aij of aij by Aij (1)i j det(M ij ) Chapter 2---Section 1 The Determinants of a Matrix ★Definition The determinant of an nxn matrix A, denoted det(A), is a scalar associated with the matrix A that is defined inductively as follows: if n 1 a11 det( A) a11 A11 a21 A21 an1 An1 if n 1 (expansion along the first column of A) where Aj1 (1)1 j det(M j1 ) , j=1, 2, …, n are the cofactors associated with the entries in the first column of A. 1.2 Determinants of Triangular matrices Theorem Let A be a lower triangular matrix (or, upper triangular matrix). Then, det(A) is the product of the diagonal elements of A. Proof If A is upper triangular, then it is easy to prove by induction. If A is lower triangular, then det( A) a11 A11 a21 A21 an1 An1 = n a i 1 i1 Ai1 It is sufficient to show that Aj1 0 for j>1. Notice that A1 j is still upper triangular and has one row of all zeros, and the diagonal contains at least one element of zero. By induction, the statement is true. * * * * 0 0 0 * 0 0 * * 0 * * * Det(I)=1. If A is triangular, then det( AT ) det( A) Chapter 2---Section 2 Properties of Determinants §2. Properties of Determinants New words and phrases Property 性质 Swap 交换 Induction 归纳法 Invertibility 可逆性 2.1 Effects of Row Operations on the Values of the Determinants (1) Row Operations of Type I: Theorem Suppose that E is an elementary matrix of type I, Then det(EA)=det(E)det(A)=-det(A), det( ET ) det( E ) =-1, det( E 1 ) 1/ det( E ) Proof Step 1: Prove the theorem under the assumption that column 1 is swapped with column k, for k>1. If this is proved, then swapping column k and column k’ will be equivalent to performing three swaps: first swapping column 1 and column k, then swapping column 1 and column k’, and finally swapping column 1 and column k. The proof is by induction on n. The base case n=1 is completely trivial.( Or, if you prefer, you may take n=2 to be the base case, and the Chapter 2---Section 2 Properties of Determinants theorem is easily proven using the formula for the determinant of a 2x2 matrix.) Suppose that the statement is true for (n-1)x(n-1) matrices. det( A) a11 A11 ak1 A21 an1 An1 = det( B) b11 B11 bk1 Bk1 bn1Bn1 = ak 1 a21 B=EA= a( k 1)1 a11 a n1 a11 a21 A= a( k 1)1 ak 1 a n1 ak 2 ak 3 a22 a23 a( k 1)2 a( k 1)3 a12 a13 an 2 an 3 a12 a13 a22 a23 a( k 1)2 a( k 1)3 ak 2 ak 3 an 2 an 3 n a i 1 i1 Ai1 n b B i 1 i1 i1 akn a2 n a( k 1) n a1n ann a1n a2 n a( k 1) n akn ann If i 1, k , then Bi1 Ai1 because M iA1 and M iB1 are the same except for two rows being swapped (交换)and bi1 ai1 . For i=1 and i=k b11 ak1 , B11 (1)11 det( M11B ) det( M11B ) ( 1) k 2 det( M kA1) ( 1) k det( M kA1) (total of k-2 swaps) b11 B11 ak1 (1) k det( M kA1 ) ak1 (1) k 1 det( M kA1 ) ak1 Ak1 , bk1 a11 bk1 Bk1 a11 (1) k 1 det( M kB1 ) a11 (1) k 1 (1) k 2 det( M 11A ) a11 det( M 11A ) a11 A11 Chapter 2---Section 2 Properties of Determinants Corollary If two rows of A are equal, then det(A)=0 (2) Row Operations of Type II Theorem If E is an elementary matrix of type II, then det(EA)=det(E)det(A)= det(A), det( E T ) det( E ) det( E 1 ) 1/ det( E ) , Proof Use induction (归纳法)on n. If B=EA, then M iB1 aM iA1 for k is not equal to k. bi1 Bi1 bi1 (1)i 1 det( M iB1 ) ai1 (1)i 1 det( M iA1 ) ai1 Ai1 For i=k, bk1Bk1 ai1 (1)i 1 det( M kB1 ) ai1 (1)i 1 det( M kA1 ) ai1 Ak1 Corollary If A has a row of all zeros, then det(A)=0 (3) Row Operations of Type III Theorem Suppose that Chapter 2---Section 2 Properties of Determinants a11 ai1 C= ak1 a 'k1 a n1 a12 a13 ai 2 ai 3 ak 2 a 'k 2 a k 3 a 'k 3 an 2 an 3 a11 ai1 A= ak 1 a n1 a11 ai1 B= a 'k 1 a n1 ain akn a 'kn ann a1n a12 a13 ai 2 ai 3 ak 2 ak 3 an 2 an 3 a12 a13 ai 2 ai 3 a 'k 2 a 'k 3 an 2 an 3 a1n ain akn ann a1n ain a 'kn ann Then det(C)=det(A)+det(B) Proof: Use induction to show that ci1Ci1 ai1 Ai1 bi1Bi1 for i=1, 2, …, n. Theorem If E is an elementary matrix of type III, then det(EA)=det(E)det(A)=det(A), det( E ) det( E T ) 1 , det( E 1 ) 1/ det( E ) Chapter 2---Section 2 Properties of Determinants Corollary If two rows of A are proportional, then det(A)=0. 2.2 Determinants and invertibility Theorem A is nonsingular if and only if det(A) is not zero. Proof A Ek E2 E1U . Use induction to show that det( A) det( Ek ) det( E2 ) det( E1 ) det(U ) , where U is upper triangular. det(A) is zero if and only if its reduced echelon form U has a determinant of zero. If A is singular, then U must have a row of all zeros and hence det(U)=0(U is also triangular). If A is nonsingular, then U=I, det(A) is not zero. 2.3 Determinants of Products of Matrices Theorem det(AB)=det(A)det(B) Proof: If det(B)=0, then B is singular, then AB is also singular. So both sides are equal. If det(A)=0 and det(B) is not zero, then AY=0 has a nontrivial solution Y. ABX=0 has a nontrivial solution X B1Y , so AB is singular. det(AB)=0. If det(A) is not zero, the A is row equivalent to I. A Ek E2 E1 . Chapter 2---Section 2 Properties of Determinants Then by the result above, the theorem is proven. 2.4 Determinants of Transposes Theorem det( A) det( AT ) Proof Express A in row echelon form U. A Ek E2 E1U , where U is upper triangular and E’s are elementary matrices. AT U T E1T E2T Ek T For triangular matrices and elementary matrices B, we have det( B) det( BT ) 2.5 Effects of Column Operations on the Values of the Determinants Since det(AE)=det(A)det(E), we have (1) Interchanging two columns of a matrix changes the sign of the determinant. (2) Multiplying a single column of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar. (3) Adding a multiple of one column to another does not change the value of the determinant. Chapter 2---Section 2 Properties of Determinants 2.6 Cofactor expansion along any column or row Theorem 2.1.1 If A is an nxn matrix with n 2 , then det(A) can be expressed as a cofactor expansion using any row or column of A. Proof: Let a11 ai1 A= ak 1 a n1 a1 j aij B= akj a nj a12 a13 a1 j ai 2 ai 3 aij ak 2 ak 3 akj an 2 an 3 anj a12 a13 a11 ai 2 ai 3 ai1 ak 2 ak 3 ak 1 an 2 an 3 an1 a1n ain akn ann a1n ain akn ann Then det(A)=-det(B) n det( B) bi1 (1)1i det( M iB1 ) i 1 n aij (1)1i det( M iB1 ) (compare i 1 between M iB1 and M ijA ) n aij (1)1i (1) j 2 det( M ijA ) i 1 n n i 1 i 1 aij (1)i j det( M ijA ) aij Aij the difference Chapter 2---Section 2 Properties of Determinants n Hence, det( A) det( B) aij Aij . i 1 This proves that det(A) can be expanded along any column of A. Since det( A) det( AT ) , det(A) can be expanded along any row of A. Assignment Hand in: 11, 12, 15, 16, 17, Not required: 19, 20, Chapter 2---Section 2 Properties of Determinants §3. Cramer’s Rule New words and phrase Cramer’s rule 克莱姆法则 Adjoint matrix 伴随矩阵 Expand 展开 In this section, we learn a method for computing the inverse of a nonsingular matrix A using determinants. We also learn a method for solving AX=B using determinants. Both methods are dependent on the following lemma. Lemma 2.2.1 Let A be an nxn matrix. If A jk denotes the cofactor of a jk , then n a k 1 ik Ajk det( A) ij n and a k 1 ki Akj det( A) ij Proof If i=j, then it is obvious. If i is not equal to j, then consider a matrix obtained by replacing the jth row of A by the ith row of A and expand along the jth row. Chapter 2---Section 2 Properties of Determinants a11 ai1 B= ai1 a n1 a12 a13 a1k ai 2 ai 3 aik ai 2 ai 3 aik an 2 an 3 ank a1n ain , det(B)=0 ain ann Let A be an nxn matrix, we define a new matrix called the adjoint of A A11 Ai1 adjA= Ak1 A n1 A12 A13 A1 j Ai 2 Ai 3 Aij Ak 2 Ak 3 Akj An 2 An 3 Anj T A1n A11 Ain = Akn A1 j Ann A1n A21 A31 A2 j A3 j A2 n A3n An1 Anj Ann To form the adjoint, we must replace each term by its cofactor and then transpose the resulting matrix. By the lemma above, we obtain A(adjA)=(adjA)A=det(A)I If A is nonsingular, det(A) is not zero, thus, A1 1 adjA det( A) Theorem 2.3.1 Let A be an nxn nonsingular matrix, an let b Rn . Let A j be the matrix obtained by replacing the jth column of A by b. If x is the unique solution to AX=b, then xj det( Aj ) det( A) Chapter 2---Section 2 Properties of Determinants Cramer’s rule gives a convenient method for expressing the solution to an nxn system of linear equations in terms of determinants. Proof X A1B a11 ai1 Aj ak1 a n1 1 (adjA) B det( A) a12 a13 b1 ai 2 ai 3 bi ak 2 ak 3 bk an 2 an 3 bn a1n ain n bs Asj A1 j s 1 akn ann A2 j A Summary on determinants Definition of determinants Determinants of triangular matrices Effects of row operations on determinants Determinants and Nonsingularity Determinants of products of matrices Determinants of transposes Effects of column operations on the values of determinants Expansion along any row or column Cramer’s rule Another way to calculate the inverse of a matrix b1 b Anj 2 bn Chapter 2---Section 2 Properties of Determinants Assignment Hand in: 6, 7, 8, 10, 11, 12, 13 Not required: 14