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Unformatted text preview: McGill University Math 236: Algebra 2 Assignment 4: due Friday, March 16, 2006 1. (Bonus Question) Let A = • a b c d ‚ . (a) Show that A 2- ( a + d ) A + ( ad- bc ) I = 0. (b) Show that x 2- ( a + d ) x + ( ad- bc ) is the minimal polynomial of A if A 6 = λI for any scalar λ . (c) Show that the minimal polynomial of A is ( x- λ ) 2 if and only if A is similar to • λ 1 λ ‚ . 2. Two companies A and B compete for market share of their product. Suppose that that the first month they both have the market share but that after the second month half of the purchasers of A’s product switched to B’s product and 1 / 3 of the purchasers of B’s product switch ed to A’s product. If x n ,y n are the market shares for A’s and B’s product respectively after n months and the above trend continues, show that • x n +1 y n +1 ‚ = • 1 2 1 3 1 2 2 3 ‚• x n y n ‚ . Find the limiting market shares, i.e., find lim x n , lim y n as n → ∞ . Hint: Look at the following bonus...
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This homework help was uploaded on 04/18/2008 for the course MATH 236 taught by Professor Toth during the Winter '06 term at McGill.
- Winter '06