Final Exam 2 - Answer Key to Final Examination Version 2...

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Answer Key to Final Examination, Version 2, Math. 33A, Spring Quarter, 2003 1. Which of the following four sets is a field? Subset yes or no reason The rational numbers . Yes One can add, substract, multiply and divide in . a : a in with a 0 No 1 is not in the set. a b b a : a and b are in Yes By computation, the set is closed under addition, substraction and multiplication, and one can do division if ab 0 a b 0 a : a and b are in No 0 1 0 0 does not have inverse. 2 . Answer the following questions: (a) Find all values of k when 1 1 1 1 2 k 1 4 k 2 is not invertible. We need to find when the rank of the matrix is less than 3. By row operation we get 1 1 1 1 2 k 1 4 k 2 2nd 1st, 3rd 2nd 1 1 1 0 1 k 1 0 2 k k 1 1st 2nd, 3rd 2 2nd 1 0 2 k 0 1 k 1 0 0 k 2 k 1 From this, the matrix is invertible if either k 1 or k 2. (b) Compute the determinant of 1 1 1 1 2 k 1 4 k 2 . By the above row operation, we find that the determinant is k 1 k 2 . (c) When 1 1 1 1 2 k 1 4 k 2 is not invertible, find all linearly independent vectors among column vectors of 1 1 1 1 2 k 1 4 k 2 ( Each redundant vector given costs 10 points ).
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