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Unformatted text preview: THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear Mathematics 2012 Tutorial 6 1. Define a sequence of numbers { u n  n ≥ } by u = 0 , u 1 = 1 , u 2 = 2 and for all n ≥ , u n +3 = u n +2 + 1 4 u n +1 1 4 u n . a ) Express the recurrence relation for u n in matrix form. b ) Find a formula for u n in terms of n . c ) Find lim n →∞ u n . 2. Let L = b 1 b 2 b 3 s 1 s 2 be the Leslie matrix for an animal population with 3 age groups. a ) If λ is an eigenvalue of L show that λ 3 = b 1 λ 2 + s 1 b 2 λ + s 1 s 2 b 3 . b ) Verify that u = 1 s 1 λ s 1 s 2 λ 2 is an eigenvector of L corresponding to eigenvalue λ . c ) Use parts a) and b) to find the unique positive eigenvalue, and a corresponding eigenvector, for the following Leslie matrix M = 0 4 4 3 4 0 0 2 3 . 3. At any time t , the populations of two species in symbiotic relationship (each population support ing the other) are denoted by x ( t ) and y ( t ) . They are given by the system of linked differential....
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This note was uploaded on 02/06/2012 for the course MATH 2061 taught by Professor Notsure during the Three '09 term at University of Sydney.
 Three '09
 NOTSURE
 Math

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