A_PMT_1_M2J_F09 - Math 2J Lecture B Fall 2009(Instructor...

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Unformatted text preview: Math 2J Lecture B Fall 2009 (Instructor: Professor R. C. Reilly) Answer Key for Practice Midterm #1 (1) (a) Let I 3 denote the 3-by-3 identity matrix. We use Gauss-Jordan Elimination on the matrix [ A | I 3 ] to obtain A- 1 . You should be able to figure out the row operations being used, so I won’t list them out here. (I do list them out in some of the other problems.) [ A | I 3 ] = 2- 1 4 1 0 0 6 0 0 1 0- 1 2 3 0 0 1 → - 1 2 3 0 0 1 6 0 0 1 0 2- 1 4 1 0 0 → - 1 2 3 0 0 1 0 6 0 1 0 0 3 10 1 0 2 → - 1 2 3 0 0 1 0 3 10 1 0 2 0 6 0 1 0 → - 1 2 3 0 0 1 0 3 10 1 0 2 0 0- 20- 2 1- 4 → - 1 2 3 1 0 3 10 1 2 0 0 1 1 / 10- 1 / 20 1 / 5 → - 1 2 0- 3 / 10 3 / 20 2 / 5 0 3 0 1 / 2 0 0 1 1 / 10- 1 / 20 1 / 5 → - 1 2 0- 3 / 10 3 / 20 2 / 5 0 1 0 1 / 6 0 0 1 1 / 10- 1 / 20 1 / 5 → - 1 0 0- 3 / 10- 11 / 60 2 / 5 0 1 0 1 / 6 0 0 1 1 / 10- 1 / 20 1 / 5 → 1 0 0 3 / 10 11 / 60- 2 / 5 0 1 0 1 / 6 0 0 1 1 / 10- 1 / 20 1 / 5 From this one reads off A- 1 = 3 / 10 11 / 60- 2 / 5 1 / 6 1 / 10- 1 / 20 1 / 5 (b) We learned in the lecture that when A- 1 exists, then the solution to A x = b is A- 1 b . In this case that means x = 3 / 10 11 / 60- 2 / 5 1 / 6 1 / 10- 1 / 20 1 / 5 3- 1 = 43 / 60- 1 / 6 7 / 20 (2) Let us transform the augmented matrix C = [ A | b ] =- 1 2 4- 2 4 3- 6 5 1 3 to reduced row-echelon form by the usual elementary row operations:- 1 2 4- 2 4 3- 6 5 1 3 (1) →- 1 2 4- 2 4 0 0 17- 5 15 (2) →- 1 2 4- 2 4 0 0 1- 5 / 17 15 / 17 (3) →- 1 2 0- 14 / 17 8 / 17 0 0 1- 5 / 17 15 / 17 (4) → 1- 2 0 14 / 17- 8 / 17 0 1- 5 / 17 15 / 17 The steps in the preceding row-reduction can be summarized as follows: Step (1) : R 2 → R 2 + 3 R 1 . Note: By the end of this step it is clear that the original system is consistent and that x 1 and x 3 are the lead variables, hence x 2 and x 4 are the free variables. Step (2) : R 2 → R 2 / 17 Step (3) : R 1 = R 1- 4 R 2 . This is the first part of the ‘Jordan’ half of ’Gauss-Jordan Elimination’. Step (4) : R 1 → (- 1) R 1 Note: This completes the ‘Jordan’ half; the final matrix is in reduced row-echelon form....
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A_PMT_1_M2J_F09 - Math 2J Lecture B Fall 2009(Instructor...

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