Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 06 (a)
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Question B 06 (a) 

Consider the threecomponent differential equation . The matrix has real entries. It has an eigenvalue and an eigenvalue with corresponding eigenvectors and (a) Write the general solution to the differential equation. 
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 

Suppose a matrix has real entries, and is an eigenvalue of with associated eigenvector . Then their complex conjugates and are also an eigenvalue and eigenvector pair of the matrix . 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. One eigenvalue and eigenvector pair, and is real. The other eigenvalue and eigenvector pair, and , is complex. Because the matrix has real entries, the complex conjugates of and must also be an eigenvalue and eigenvector pair, and . Since the matrix is a matrix, it can have at most 3 eigenvalues. So we have found all of them and they are distinct. In this case, the general solution to the differential equation is given by
Substituting the eigenvalues and eigenvectors that we found, the general solution to the differential equation is
To put this in real form, we replace the complex term and its conjugate by the real and imaginary parts of the former. Since and we then have that the general solution in real form is:
