MATH152 April 2016
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Question B 06 (a)
Consider the three-component differential equation . The matrix has real entries. It has an eigenvalue and an eigenvalue with corresponding eigenvectors
(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!
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 .
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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