Practice Midterm key

Practice Midterm key - Math 4 Winter 2008(Instructor...

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Unformatted text preview: Math 4 Winter 2008 (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 6 1- 1 2 3 1 → - 1 2 3 1 6 1 2- 1 4 1 → - 1 2 3 1 6 1 3 10 1 2 → - 1 2 3 1 3 10 1 2 6 1 → - 1 2 3 1 3 10 1 2- 20- 2 1- 4 → - 1 2 3 1 3 10 1 2 1 1 / 10- 1 / 20 1 / 5 → - 1 2- 3 / 10 3 / 20 2 / 5 3 1 / 2 1 1 / 10- 1 / 20 1 / 5 → - 1 2- 3 / 10 3 / 20 2 / 5 1 1 / 6 1 1 / 10- 1 / 20 1 / 5 → - 1- 3 / 10- 11 / 60 2 / 5 1 1 / 6 1 1 / 10- 1 / 20 1 / 5 → 1 3 / 10 11 / 60- 2 / 5 1 1 / 6 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 17- 5 15 (2) →- 1 2 4- 2 4 1- 5 / 17 15 / 17 (3) →- 1 2- 14 / 17 8 / 17 1- 5 / 17 15 / 17 (4) → 1- 2 14 / 17- 8 / 17 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 there will be two free variables in the final solution, namely x 2 and x 4 . 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’....
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This note was uploaded on 04/25/2008 for the course MATH 4 taught by Professor Reily during the Winter '08 term at UC Irvine.

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Practice Midterm key - Math 4 Winter 2008(Instructor...

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