MAP 4305 Exam Practice 1 - Practice exam I MAP 4305 Name...

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Practice exam I: MAP 4305 * September 11, 2006. Name : Student ID : This is a closed book exam and the use of calculators is not allowed. 1. Let A = parenleftbigg 1 2 2 1 parenrightbigg . (a) Calculate A 231 . (b) Discuss stability for the system x ( t + 1) = Ax ( t ). 2. Consider a network with two nodes (call them 1 and 2) and assume that there is a link from 1 to 2 (but not back), and that both 1 and 2 have a self-link. Calculate the pagerank for both nodes. (To determine the stochastic matrix, use the same rule we used on the HW problem). 3. Suppose that the vector functions parenleftbigg x 1 ( t ) x 2 ( t ) parenrightbigg and parenleftbigg y 1 ( t ) y 2 ( t ) parenrightbigg , where t R , are solutions of ˙ x = parenleftbigg a b c d parenrightbigg x. Let X ( t ) be defined as:
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