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Chapter 3 Determinants and Cramer’s Rule Matrix: an array of numbers Determinant: a number associated with each matrix. Consider matrix A: A= 5 3 2 1 The determinant of matrix A is notated in the following way: det a or A How is this notation different than regular matrix notation? Determinant of a 2 x 2 matrix: a det c b a d c b =______________________ d Find the determinant of the below matrices: 1.) 6 2 1 7 2.) 1/ 2 4 2 8 Determinant of a 3 x 3 matrix: ALWAYS MULTIPLY DIAGONALS!!!!! a det d g b e h c f i Find the determinant of the following matrix: 1.) Chapter 3 Determinants and Cramer’s Rule Cramer’s Rule and Systems of Equations Swiss Mathematician, Gabriel Cramer (1704-1752) developed a way to use matrices to solve systems of equations. His method has been named “Cramer’s Rule.” Cramer’s Rule for a 2 x 2 System: (2 equations, 2 variables) ax by c cd dy f 6x 8 y 4 4 x 5 y 4 Let A be the coefficient Matrix of the linear system: General: A Specific Case: det A: A det A: If det A 0 , then the systems has exactly one solution. The solution is: General: x y Specific Case: det A det A Ordered Pair Solution: _______________________ Chapter 3 Determinants and Cramer’s Rule Cramer’s Rule for a 3 x 3 System: (3 equations, 3 variables) General: ax by cz j Specific: dx ey fz k gx hy iz l x 2 y 3 z 2 x y z 1 3 x 4 y 4 z 4 Coefficient Matrix: Coefficient Matrix: Determinant from Calculator: _____ Let A be the coefficient matrix of the linear system. If det A 0 , then the system has exactly one solution. The solution is: General: Specific: x y z det A det A det A Ordered triple Solution:_____________________ Chapter 3 Determinants and Cramer’s Rule Use Cramer’s Rule to solve the linear system. Show your setup, but you may do the work on your calculator. 1. 2x y 3 5x 6 y 4 x 2y z 9 2. x y z 3 5 x 2 z 1 Word Problem: Science Connection: The atomic weights of three compounds are shown. Use a linear system and Cramer’s rule to find the atomic weights of carbon (C) , and hydrogen ( H ), and oxygen ( O ). HW: Determinants of Matrices: pg. 218-219 #12-20 even, 22, 25 Cramer’s Rule: pg. 218-219 #39, 41, 51,52, 60 Chapter 3 Determinants and Cramer’s Rule To Do on Graphing Calc: 1.) Hit 2nd x 1 (matrix) Scroll Over to Edit 7.) Hit enter again 2.) Hit Enter and Enter the 2X2 8.) Verify the 2 X2 by hand 3.) Now hit 2nd Quit to get back to the home screen 4.) Hit 2nd x 1 (matrix) Scroll Over to math Highlight 1:det ( 5.) Hit enter 6.) To get the matrix into home screen hit 2nd x 1 (matrix) highlight 1: [A] and enter Chapter 3 Determinants and Cramer’s Rule