A what is the rank of the matrix of coefficients

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a ) What is the rank of the matrix of coefficients? Answer: The rank is 2. This is clear once we row-reduce the augmented matrix: 1 1 1 1 1 1 - 1 1 1 1 2 - 3 2 2 2 (2) - (1) , (3) - 2(1) ----------→ 1 1 1 1 1 0 - 2 0 0 0 0 - 5 0 0 0 (3) - 5 / 2(2) -------→ 1 1 1 1 1 0 - 2 0 0 0 0 0 0 0 0 b ) Does this system have any solutions? Answer: Yes. c ) If the system has one or more solutions, describe them. Is the solution unique? Answer: Any vector of the form (1 - y - z, 0 , y, z ) T will satifisy the system. The solution is not unique.
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MATHEMATICAL ECONOMICS EXAM #1, SEPTEMBER 26, 2011 Page 2 3. Consider the set T = { x R 3 : k x k = 1 } ⊂ R 3 . a ) Is the set T open, closed, or neither when considered as a subset of R 3 ? Answer: The set is closed. b ) Prove your answer in part (a). Answer: One way to prove this is to note that f ( x ) = k x k is continuous, so if x n x , then k x n k → k vx k . Since k x n k = 1, k x k = 1 and x T . If you don’t know that the norm is continuous, use the triangle inequality to show k x n k - k x k ≤ k x n - x k , which can be used to show that the norm is continuous.
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