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pset2sol - 18.06 Problem Set 2 Solutions Problem 1 Do...

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18.06 Problem Set 2 Solutions Problem 1: Do problem 27 from section 2.5 in the book. Solution (8pts) 1 0 0 1 0 0 2 1 3 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 3 - 2 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 - 2 1 - 3 0 0 1 0 0 1 . Hence for A = 1 0 0 2 1 3 0 0 1 , A - 1 = 1 0 0 - 2 1 - 3 0 0 1 . And 1 1 1 1 0 0 1 2 2 0 1 0 1 2 3 0 0 1 1 1 1 1 0 0 0 1 1 - 1 1 0 0 1 2 - 1 0 1 1 1 1 1 0 0 0 1 1 - 1 1 0 0 0 1 0 - 1 1 1 1 1 1 0 0 0 1 0 - 1 2 - 1 0 0 1 0 - 1 1 1 0 0 2 - 1 0 0 1 0 - 1 2 - 1 0 0 1 0 - 1 1 . Hence for A = 1 1 1 1 2 2 1 2 3 , A - 1 = 2 - 1 0 - 1 2 - 1 0 - 1 1 . Problem 2: Do problem 13 from section 2.6. Solution (10pts) The required eliminations are, 1) subtracting the first row from the other rows, 2) subtracting the second row from the third and fourth, and 3) subtracting the third row from the fourth. The resulting matrix U is a a a a 0 b - a b - a b - a 0 0 c - b c - b 0 0 0 d - c . Hence, A = a a a a a b b b a b c c a b c d = 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1
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