Sample Midterm with Solutions

Sample Midterm with Solutions - March 6, 2009 MATH 270:...

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Unformatted text preview: March 6, 2009 MATH 270: Midterm Time: 2 hours 1. Consider the matrix A = 1- 1 2- 1- 1- 1 2- 1 2- 2 2 1 (a) [5 marks] Find the LU-factorization of A . (b) [5 marks] Solve the system 1- 1 2- 1- 1- 1 2- 1 2- 2 2 1 x 1 x 2 x 3 x 4 = 1 1 1 1 using the LU-factorization obtained in (a). Solution. (a) We apply elementary row operations to bring A to an upper-triangular form. 1- 1 2- 1- 1- 1 2- 1 2- 2 2 1 r 3 r 3 + r 1- 1- 1 2- 1- 1 2- 2 2- 2 2 1 r 3 r 3- r 2- 1- 1 2- 1- 1- 1 3- 2 2 1 r 4 r 4 + r 2- 1 0- 1 0 2- 1- 1 0 0- 1 3 0 0 1 r 4 r 4 + r 3- 1 0- 1 0 2- 1- 1 0 0- 1 3 0 0 3 = U The corresponding elementary matrices (in the same order as the operations) are: E 1 = 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 , E 2 = 1 0 0 0 1 0 0- 1 1 0 0 0 1 , E 3 = 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 E 4 = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 Hence, E- 1 1 = 1 0 0 0 0 1 0 0- 1 0 1 0 0 0 0 1 , E- 1 2 = 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 , E- 1 3 = 1 0 0 0 1 0 0 0 1 0- 1 0 1 E- 1 4 = 1 0 0 0 0 1 0 0 0 0 1 0 0 0- 1 1 and L = E- 1 1 E- 1 2 E- 1 3 E-...
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Sample Midterm with Solutions - March 6, 2009 MATH 270:...

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