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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 LUfactorization 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 LUfactorization obtained in (a). Solution. (a) We apply elementary row operations to bring A to an uppertriangular 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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 Winter '09
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