quiz1_S07 solutions

Quiz1_S07 solutions - Princeton University Department of Mathematics MAT 202 – Linear Algebra with Applications QUIZ 1 – Spring 2007

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Unformatted text preview: Princeton University Department of Mathematics MAT 202 – Linear Algebra with Applications QUIZ 1 – Spring 2007 SOLUTIONS (1) Consider the system x 1 + x 2- x 3 = 1 2 x 1 + kx 2 + x 3 = 5 x 1 + x 2 + kx 3 = k + 2 (a) (10 points) For what values of k does this system have exactly one solution? (b) (10 points) Solve this system with k =- 1 . Solution: (a) We apply Gauss-Jordan elimination to the system written in matrix notation: 2 4 1 1- 1 2 k 1 1 1 k ˛ ˛ ˛ ˛ ˛ ˛ 1 5 k + 2 3 5 → 2 4 1 1- 1 k- 2 3 k + 1 ˛ ˛ ˛ ˛ ˛ ˛ 1 3 k + 1 3 5 The system has exactly one solution when the corresponding rref has three leading ones, that is, when k 6 = 2 and k 6 =- 1 . (b) We replace k by- 1 above and continue elimination: 2 4 1 1- 1- 3 3 ˛ ˛ ˛ ˛ ˛ ˛ 1 3 3 5 → 2 4 1 1- 1 1- 1 ˛ ˛ ˛ ˛ ˛ ˛ 1- 1 3 5 → 2 4 1 1- 1 ˛ ˛ ˛ ˛ ˛ ˛ 2- 1 3 5 The solutions are the triples ( x 1 , x 2 , x 3 ) satisfying x 1 = 2 x 2- x 3 =- 1 ....
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This note was uploaded on 04/28/2008 for the course MAT 202 taught by Professor Staff during the Spring '08 term at Princeton.

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Quiz1_S07 solutions - Princeton University Department of Mathematics MAT 202 – Linear Algebra with Applications QUIZ 1 – Spring 2007

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