Spring2005_2

Spring2005_2 - MATH 136 Spring 05 Midterm 2 solutions 1....

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Unformatted text preview: MATH 136 Spring 05 Midterm 2 solutions 1. Find the LU factorization of A = 2 1 2 1- 4 2- 3- 1- 4 6 5 1 . Solution: Using the LU factorization algorithm we get: 2 1 2 1 0 4 1 1 0 8 9 3 r 2 + 2 r 1 r 2 r 3 + 2 r 1 r 3 1 0 0- 2 1 0- 2 0 1 2 1 2 1 0 4 1 1 0 0 7 1 r 3- 2 r 2 r 3 1 0 0- 2 1 0- 2 2 1 Hence L = 1 0 0- 2 1 0- 2 2 1 U = 2 1 2 1 0 4 1 1 0 0 7 1 . 2. Let A = 1 2 1- 1 4 3 6 9 and v = 2- 5 . [2] a) Is v Nul A ? Multiplying Av we get: 1 2 1- 1 4 3 6 9 2- 5 = - 8 5- 24 , so v is not in Nul A . [2] b) Is v Col A ? Row reducing the augmented matrix we get: 1 2 1 2- 1 4- 5 3 6 9 1 2 1 2- 1 4- 5 0 6- 6 . Hence v Col A since the system is consistent. 1 2 [4] 3. Find the inverse of A = 1 2 1 1 3 4 2 4 1 . Solution: Row reducing the augmented matrix we get: 1 2 1 1 0 0 1 3 4 0 1 0 2 4 1 0 0 1 1 2 1 1 0 0 0 1 3- 1 1 0 0 0- 1- 2 0 1 1 2- 1 0 1 0 1- 7 1 3 0 0- 1-...
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This note was uploaded on 09/04/2009 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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Spring2005_2 - MATH 136 Spring 05 Midterm 2 solutions 1....

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