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7.5 Determinants and Cramer’s Rule • Subscript notation for the matrix A a11 a 21 A a31 am1 a12 a13 a22 a23 a32 a33 am 2 am 3 a1n a2 n a3n amn The row 1, column 1 element is a11; the row 2, column 3 element is a23; and, in general, the row i, column j element is aij. Copyright © 2011 Pearson Education, Inc. Slide 7.5-1 7.5 Determinants of 2 × 2 Matrices • Associated with every square matrix is a real number called the determinant of A. In this text, we use det A. The determinant of a 2 × 2 matrix A, where a11 a12 A a21 a22 is a real number defined as det A a11a22 a21a12 . Copyright © 2011 Pearson Education, Inc. Slide 7.5-2 7.5 Determinants of 2 × 2 Matrices Example Find det A if A 3 4. 6 8 Analytic Solution Graphing Calculator Solution det A det 3 4 6 8 3(8) 6(4) 48 Copyright © 2011 Pearson Education, Inc. Slide 7.5-3 7.5 Determinant of a 3 × 3 Matrix The determinant of a 3 × 3 matrix A, where a11 a12 a13 A a21 a22 a23 a a a 31 32 33 is a real number defined as det A a11a22 a33 a12 a23a31 a13a21a32 a31a22 a13 a32 a23a11 a33a21a12 . Copyright © 2011 Pearson Education, Inc. Slide 7.5-4 7.5 Determinant of a 3 × 3 Matrix • A method for calculating 3 × 3 determinants is found by re-arranging and factoring this formula. a11 a12 a13 det A a21 a22 a23 a a a 31 32 33 a11 (a22 a33 a32 a23 ) a21 (a12 a33 a32 a13 ) a31 (a12 a23 a22 a13 ) Each of the quantities in parentheses represents the determinant of a 2 × 2 matrix that is part of the 3 × 3 matrix remaining when the row and column of the multiplier are eliminated. Copyright © 2011 Pearson Education, Inc. Slide 7.5-5 7.5 The Minor of an Element • The determinant of each 3 × 3 matrix is called a minor of the associated element. • The symbol Mij represents the minor when the ith row and jth column are eliminated. Copyright © 2011 Pearson Education, Inc. Slide 7.5-6 7.5 The Cofactor of an Element Let Mij be the minor for element aij in an n × n matrix. The cofactor of aij, written Aij, is Aij 1 i j • M ij . To find the determinant of a 3 × 3 or larger square matrix: 1. Choose any row or column, 2. Multiply the minor of each element in that row or column by a +1 or –1, depending on whether the sum of i + j is even or odd, 3. Then, multiply each cofactor by its corresponding element in the matrix and find the sum of these products. This sum is the determinant of the matrix. Copyright © 2011 Pearson Education, Inc. Slide 7.5-7 7.5 Finding the Determinant 2 3 2 Example Evaluate det 1 4 3 , expanding 0 2 1 by the second column. Solution First find the minors of each element in the second column. M 12 det 1 1 M 22 det 2 1 M 32 det 2 1 Copyright © 2011 Pearson Education, Inc. 3 1(2) (1)( 3) 5 2 2 2(2) (1)( 2) 2 2 2 2(3) (1)( 2) 8 3 Slide 7.5-8 7.5 Finding the Determinant Now, find the cofactor. A12 (1)1 2 M 12 (1)3 (5) 5 A22 (1) 2 2 M 22 (1) 4 (2) 2 A32 (1)3 2 M 32 (1)5 (8) 8 The determinant is found by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products. 2 3 2 det 1 4 3 a12 A12 a22 A22 a32 A32 2 1 0 3(5) (4)(2) (0)(8) 23 Copyright © 2011 Pearson Education, Inc. Slide 7.5-9 7.5 Cramer’s Rule for 2 × 2 Systems The solution of the system a1 x b1 y c1 a2 x b2 y c2 , Dx is given by x D where and Dy y , D a1 c1 b1 Dx det , Dy det a2 c2 b2 Copyright © 2011 Pearson Education, Inc. c1 , c2 a1 b1 and D det 0. a2 b2 Slide 7.5-10 7.5 Applying Cramer’s Rule to a System with Two Equations Example Use Cramer’s rule to solve the system. 5x 7 y 1 6x 8 y 1 Analytic Solution By Cramer’s rule, x Dx D and y Dy D . D det 5 7 5(8) 6(7) 2 6 8 1 7 (1)(8) (1)(7) 15 Dx det 1 8 5 1 (5)(1) (6)( 1) 11 Dy det 6 1 Copyright © 2011 Pearson Education, Inc. Slide 7.5-11 7.5 Applying Cramer’s Rule to a System with Two Equations Dx 15 15 x and D 2 2 Dy 11 11 y D 2 2 The solution set is 152 , 112 . Graphing Calculator Solution Enter D, Dx, and Dy as matrices A, B, and C, respectively. Copyright © 2011 Pearson Education, Inc. Slide 7.5-12 7.5 Cramer’s Rule for 3 × 3 Systems The solution for the system a1 x b1 y c1 z d1 a2 x b2 y c2 z d 2 a3 x b3 y c3 z d3 , is given by x Dx , D y Dy D , and z z , D D where d1 Dx det d 2 d3 a1 Dz det a2 a3 Copyright © 2011 Pearson Education, Inc. b1 b2 b3 b1 b2 b3 c1 c2 , c3 a1 d1 c1 Dy det a2 d 2 c2 , a3 d3 c3 d1 a1 b1 c1 d 2 , and D det a2 b2 c2 0. a3 b3 c3 d3 Slide 7.5-13