Science:Math Exam Resources/Courses/MATH152/April 2022/Question B4 (a)
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Question B4 (a) 

Question: Consider the system of differential equations , where (a) Find the eigenvalues and corresponding eigenvectors of . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
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 

Recall that the eigenvalues solve the characteristic equation, given by . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. To compute the eigenvalues, first form the characteristic equation, then solve for its zeros. To solve for the zeros, use thus the eigenvalues are and . Since the eigenvalues are complex conjugates of each other, we know that the eigenvectors are also complex conjugates of each other, meaning we only need to compute one eigenvector to know both. In question 17 we found the eigenvectors of a matrix by solving for the components or using basic algebra. For this question we will solve that same system using row operations. Solving for , we write the system as , row operations on the augmented system of equations gives If , then we have . Choosing (arbitrarily) , we get , and the two eigenvectors are 