Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 04 (a)
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Question B 04 (a) 

Let A be the matrix given by It is known that is an eigenvalue of A. Find an eigenvector associated with the eigenvalue . 
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 

To calculate the eigenvectors v for the eigenvalue 3, find the solution of 
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. Proceeding as usual, we look for nonzero solutions to The free variable is v_{2}, while there is a system of two equations and two unknowns for v_{1} and v_{3}. The (simplest and only) solution to this smaller system is v_{1} = 0 = v_{3}. Hence, any vector of the form with is an eigenvector of A with eigenvalue 3. 