02test1 - Mathematical Economics Midterm#1 October 3 2002...

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Mathematical Economics Midterm #1, October 3, 2002 You have until 4:45 to complete this exam. Answer all five questions. Each question is worth 20 points, for a total of 100 points. Good luck! 1. Consider the vector x = 3 2 1 . Find the co-ordinates of x in the basis B = { b 1 , b 2 , b 3 } = 1 1 0 , 0 1 1 , 0 0 1 . 2. Let A = 1 6 - 7 3 1 9 - 6 4 1 3 - 8 4 . a ) What is the rank of A ? b ) Does the equation A x = b have a solution x R 4 for every b R 3 ? c ) If A x = b can be solved, how many solutions does it have? 3. The stationary distribution of the Markov employment model obeys ( q - 1) x + py = 0 (1 - q ) x - py = 0 x + y = 1 where 0 < p < 1 and 0 < q < 1. Does this system have a solution ( x, y ) 0 ? If so, is there a unique solution or multiple solutions? Finally, does the system have a non-negative solution?
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