Science:Math Exam Resources/Courses/MATH307/April 2006/Question 02 (a)
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Question 02 (a) 

Consider the diagonalizable matrix (a) Find a solution of the difference equation satisfying the initial condition . Determine the limit of the ratio as . 
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Science:Math Exam Resources/Courses/MATH307/April 2006/Question 02 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We want to find the ratio of as for the difference equation . The difference equation suggests that for every increment of 1 unit time, we want to multiply by . The matrix A is already given in the diagonalized form of Given the initial condition of , we have the equation , which can also be expressed as the form where you can obtain c_{1}, c_{2} by plugging in the initial condition x_{0} and solving As , the component with would reach 0 and you would be left with and therefore the ratio of would just be . 