#### Transcript Cramer Rule

Determinants Every n n matrix A is associated with a real number called the determinant of A, written A. The determinant of a 2 2 matrix is defined as follows. 5.3 - 1 Determinant of a 2 2 Matrix a1 1 If A = a21 A a1 1 a1 2 a21 a22 a1 2 , th e n a22 a1 1a 2 2 a 2 1a1 2 . 5.3 - 2 Note Matrices are enclosed with square brackets, while determinants are denoted with vertical bars. A matrix is an array of numbers, but its determinant is a single number. 5.3 - 3 Determinants The arrows in the following diagram will remind you which products to find when evaluating a 2 2 determinant. 5.3 - 4 Example 1 Let A 3 = 6 EVALUATING A 2 2 DETERMINANT 4 . 8 Find A. Solution Use the definition with a1 1 3, a1 2 4, a 2 1 6, a 2 2 8 . A 3 8 a11 a22 6 4 a21 a12 24 24 48 5.3 - 5 Determinant of a 3 3 Matrix If a1 1 A = a21 a 3 1 a1 2 a22 a32 a1 3 a 2 3 , th e n a 3 3 a1 1 a1 2 a1 3 A a21 a22 a 2 3 ( a 1 1a 2 2 a 3 3 a 1 2 a 2 3 a 3 1 a 1 3 a 2 1a 3 2 ) a31 a32 a 3 3 ( a 3 1a 2 2 a1 3 a 3 2 a 2 3 a1 1 a 3 3 a 2 1a1 2 ). 5.3 - 6 Evaluating The terms on the right side of the equation in the definition of A can be rearranged to get a1 1 a1 2 a1 3 A a21 a22 a 2 3 a1 1 ( a 2 2 a 3 3 a 3 2 a 2 3 ) a 2 1 ( a1 2 a 3 3 a 3 2 a1 3 ) a31 a32 a 3 3 a 3 1 ( a1 2 a 2 3 a 2 2 a 1 3 ). Each quantity in parentheses represents the determinant of a 2 2 matrix that is the part of the matrix remaining when the row and column of the multiplier are eliminated, as shown in the next slide. 5.3 - 7 Evaluating a1 1 ( a 2 2 a 3 3 a 3 2 a 2 3 ) a 2 1 ( a1 2 a 3 3 a 3 2 a1 3 ) a 3 1 ( a1 2 a 2 3 a 2 2 a1 3 ) a1 1 a21 a 3 1 a1 2 a1 1 a21 a 3 1 a1 2 a1 1 a21 a 3 1 a1 2 a22 a32 a22 a32 a22 a32 a1 3 a23 a 3 3 a1 3 a23 a 3 3 a1 3 a23 a 3 3 5.3 - 8 Cramer’s Rule for Two Equations in Two Variables Given the system a1 x b1 y c 1 a 2 x b2 y c 2 if then the system has the unique solution x where D a1 b1 a2 b2 , Dx D Dx and y c1 b1 c2 b2 Dy D , , and Dy a1 c1 a2 c2 . 5.3 - 9 Caution As indicated in the preceding box, Cramer’s rule does not apply if D = 0. When D = 0 the system is inconsistent or has infinitely many solutions. For this reason, evaluate D first. 5.3 - 10 APPLYING CRAMER’S RULE TO A 2 2 SYSTEM Use Cramer’s rule to solve the system Example 4 5 x 7 y 1 6x 8y 1 Solution Dy Dx By Cramer’s rule, and y D . Find D first, D since if D = 0, Cramer’s rule does not apply. If D ≠ 0, then find Dx and Dy. x D 5 7 6 8 5(8 ) 6(7 ) 2 Dy 5 1 6 1 Dx 1 7 1 8 1(8 ) 1(7 ) 1 5 5(1) 6( 1) 1 1 5.3 - 11 APPLYING CRAMER’S RULE TO A 2 2 SYSTEM By Cramer’s rule, Example 4 Dy 15 15 11 11 x and y . D 2 2 D 2 2 Dx 11 15 , 2 2 The solution set is as can be verified by substituting in the given system. 5.3 - 12 General form of Cramer’s Rule Let an n n system have linear equations of the form a1 x 1 a 2 x 2 a 3 x 3 a n x n b . Define D as the determinant of the n n matrix of all coefficients of the variables. Define Dx1 as the determinant obtained from D by replacing the entries in column 1 of D with the constants of the system. Define Dxi as the determinant obtained from D by replacing the entries in column i with the constants of the system. If D 0, the unique solution of the system is x1 D x1 D , x2 Dx2 D , x3 Dx3 D , , xn D xn D . 5.3 - 13 APPLYING CRAMER’S RULE TO A 3 3 SYSTEM Use Cramer’s rule to solve the system. x y z20 2x y z 5 0 x 2y 3z 4 0 Example 5 Solution x 2x y z 2 y z 5 x 2y 3z 4 Rewrite each equation in the form ax + by + cz + = k. 5.3 - 14 APPLYING CRAMER’S RULE TO A 3 3 SYSTEM Verify that the required determinants are Example 5 1 2 1 1 1 1 D 2 1 1 3, D x 5 1 1 7, 1 2 3 2 3 1 2 Dy 2 5 1 4 1 4 1 1 1 2 2, D z 2 1 3 2 1 2 5 21. 4 5.3 - 15 APPLYING CRAMER’S RULE TO A 3 3 SYSTEM Example 5 Thus, x Dx D 7 3 7 3 and z , Dz D y Dy D 21 3 so the solution set is 22 3 22 3 , 7, 7 22 , 7 . , 3 3 5.3 - 16 Note When D = 0, the system is either inconsistent or has infinitely many solutions. Use the elimination method to tell which is the case. Verify that the system in Example 6 is inconsistent, so the solution set is ø. 5.3 - 17