# 01samp1 - Mathematical Economics Midterm#1 October 2 2000...

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Mathematical Economics Midterm #1, October 2, 2000 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. (Linear Systems) Consider the linear system x + y + z = 13 x + 5 y + 10 z = 61 . Show that this system has solutions. How many solutions are there? Can you find a solution where x , y , and z are all positive integers? 2. (Matrices) Let A and B be n × n matrices. Show that if ( A + B ) 2 = A 2 + 2 AB + B 2 , then AB = BA . 3. (Determinant) Consider the matrix A = 3 - 4 - 1 0 Find all λ so that det( A - λI ) = 0. 4. (Linear Systems) Demand for good 1 is e 1 - ap 1 + bp 2 ; demand for good 2 is e 2 + cp 1 - dp 2 ; the supply of good i is s i . Here a , b , c , d , e i , and s

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Unformatted text preview: are all positive, and s i > e i . a ) What system oF equations do you get when you set supply equal to demand in both markets? b ) What criterion must be met in order to solve For p 1 and p 2 ? c ) What additional conditions must be satisfed in order to get positive equilibrium prices p i ? 5. (Bases) Suppose x = (1 , 3 , 7) T . Consider the basis B = { b 1 , b 2 , b 3 } where b 1 = (1 , 1 , 0) T , b 2 = (0 , 2 , 4) T , and b 3 = (0 , 3 , 9) T . ±ind the coordinates oF x in the basis B (i.e., fnd α 1 , α 2 , and α 3 with α 1 b 1 + α 2 b 2 + α 3 b 3 = x ....
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