MidtermMath2JSummer11_solutions

MidtermMath2JSummer11_solutions - Math 2J Summer 11...

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Unformatted text preview: Math 2J, Summer 11 Instructor : Yunho Kim The Midterm Exam Date : Aug. 17th UID Name Class Lec. B, 9:00am - 10:50am Number Score 1. a) b) 2. 3. a) b) 4. 5. a) b) Total Score / 100 1 Problem 1. Solve the following systems of linear equations. a. (15pts) braceleftBigg x 1 − 2 x 2 = 5 4 x 1 + x 2 = − 7 Solution. ( x 1 , x 2 ) = ( − 1 , − 3) . b. (15pts) 2 x 1 − x 2 − 3 x 3 = − x 1 + 4 x 2 + 2 x 3 = − 3 5 x 1 − 3 x 2 − 2 x 3 = 6 Solution. ( x 1 , x 2 , x 3 ) = (1 , − 1 , 1) . 2 Problem 2. (15pts) Let a 2 × 2 matrix A be defined by A = − 2 3 1 1 . Find elementary matrices E 1 , E 2 , . . ., E k such that A = E k E k- 1 ··· E 1 . Solution. The answer is not unique. My choice is that if F 1 = 1 1 , F 2 = 1 2 1 , F 3 = 1 1 / 5 , F 4 = 1 − 1 1 , then F 4 F 3 F 2 F 1 A = I , that is, A = F- 1 1 F- 1 2 F- 1 3 F- 1 4 . Therefore, E 1 = F- 1 4 = 1 1 1 , E 2 = F- 1 3 = 1 5 , E 3 = F- 1 2 =...
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MidtermMath2JSummer11_solutions - Math 2J Summer 11...

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