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Transcript
Chapter 2 • Determinants by Cofactor Expansion • Evaluating Determinants by Row Reduction • Properties of the Determinants, Cramer’s Rule 4/30/2017 1 Minor and Cofactor a b Let A be a 2x2 matrix A c d Then ad-bc is called the determinant of the matrix A, and is denoted by The symbol det(A). We will extend the concept of a determinant to square matrices of all orders. 4/30/2017 2 Minor and Cofactor Definition If A is a square matrix, then • The minor of entry aij, denoted Mij, is defined to be the determinant of the submatrix that remains after the ith row and jth column are deleted from A. • The number (-1)i+j Mij is denoted by Cij and is called the cofactor of entry aij Remark Note that Cij = Mij and the signs checkerboard pattern: (1)i j in the definition of cofactor form a 4/30/2017 ... ... ... ... 3 3 Example: Let A 2 1 1 4 5 6 . Find the minor and cofactor of a12, and a23 4 8 Solution: 4/30/2017 4 Cofactor Expansion Theorem 2.1.1 (Expansions by Cofactors) The determinant of an nn matrix A can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products; that is, for each 1 i, j n det(A) = a1jC1j + a2jC2j +… + anjCnj (cofactor expansion along the jth column) and det(A) = ai1Ci1 + ai2Ci2 +… + ainCin (cofactor expansion along the ith row) 4/30/2017 5 3 Example: Let A 2 5 1 0 4 3 4 2 . Evaluate det (A) by cofactor expansion along the first column of A. Solution: 4/30/2017 6 Determinant of an Upper Triangular Matrix Theorem If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then det(A) is the product of the entries on the main diagonal of the matrix; that is, det(A)=a11a22…ann. Arrow Technique for determinants of 2x2 and 3x3 matrices. a11 a12 a21 a22 a11 a12 a21 a22 a31 a32 4/30/2017 a11a22 a12 a21 a13 a23 a11a22 a33 a12 a23a31 a13a21a32 a13a22 a31 a12 a21a33 a11a23a32 a33 7 Section 2.3 Properties of Determinants; Cramer’s Rule Definition (Adjoint of a Matrix) If A is any nn matrix and Cij is the cofactor of aij, then the matrix C11 C12 C1n C C C 2n 21 22 C C C nn n1 n 2 is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A and is denoted by adj(A) 4/30/2017 8 3 2 1 6 3 . Find the adjoint of A. Example: Let A 1 2 4 0 Solution: 4/30/2017 9 Theorems Theorem 2.1.2 (Inverse of a Matrix using its Adjoint) If A is an invertible matrix, then A1 1 adj( A) det( A) 3 2 1 Example: Use theorem 2.1.2. to find the inverse of A 1 6 3 2 4 0 Solution: 4/30/2017 10 Cramer’s Rule Theorem 2.1.4 (Cramer’s Rule) If Ax = b is a system of n linear equations in n unknowns such that det(A) 0 , then the system has a unique solution. This solution is x1 det( An ) det( A1 ) det( A2 ) , x2 ,, xn det( A) det( A) det( A) where Aj is the matrix obtained by replacing the entries in the jth column of A by the entries in the matrix b1 b b 2 bn 4/30/2017 11 Example: Use Cramer’s rule to solve x1 2 x3 6 3x1 4 x2 6 x3 30 x1 2 x2 3x3 8 Solution: 4/30/2017 12 Properties of the Determinant Function Suppose that A and B are nxn matrices and k is any scalar. We obtain det(kA) k n det( A) But There is no simple relationship exists between det(A), det(B), and det(A+B) in general. In particular, det(A+B) is usually not equal to det(A) + det(B). Example: Consider 4/30/2017 1 2 3 1 4 3 A , B , A B 1 3 3 8 2 5 13 Properties of the Determinants Sum Theorem 2.3.1 Let A, B, and C be nn matrices that differ only in a single row, say the r-th, and assume that the r-th row of C can be obtained by adding corresponding entries in the r-th rows of A and B. Then det(C) = det(A) + det(B) The same result holds for columns. Example 1 det 2 1 0 4/30/2017 7 5 1 0 3 det 2 1 4 1 7 (1) 7 5 1 0 3 det 2 0 4 7 7 5 0 3 1 1 14 Properties of the Determinants Multiplication Lemma 2.3.2 If B is an nn matrix and E is an nn elementary matrix, then det(EB) = det(E) det(B) Theorem 2.3.3 (Determinant Test for Invertibility) A square matrix A is invertible if and only if det(A) 0 4/30/2017 15 Theorem 2.3.4 If A and B are square matrices of the same size, then det(AB) = det(A) det(B) Theorem 2.3.5 1 If A is invertible, then det( A ) 4/30/2017 1 det( A) 16 Theorem 2.3.6 (Equivalent Statements) If A is an nn matrix, then the following are equivalent • A is invertible. • Ax = 0 has only the trivial solution • The reduced row-echelon form of A as In • A is expressible as a product of elementary matrices • Ax = b is consistent for every n1 matrix b • Ax = b has exactly one solution for every n1 matrix b • det(A) 0 4/30/2017 17