Science:Math Exam Resources/Courses/MATH152/April 2011/Question B 03 (a)
• QA 1 • QA 2 • QA 3 • QA 4 • QA 5 • QA 6 • QA 7 • QA 8 • QA 9 • QA 10 • QA 11 • QA 12 • QA 13 • QA 14 • QA 15 • QA 16 • QA 17 • QA 18 • QA 19 • QA 20 • QA 21 • QA 22 • QA 23 • QA 24 • QA 25 • QA 26 • QA 27 • QA 28 • QA 29 • QA 30 • QB 1(a) • QB 1(b) • QB 1(c) • QB 2(a) • QB 2(b) • QB 3(a) • QB 3(b) • QB 3(c) • QB 4(a) • QB 4(b) • QB 4(c) • QB 4(d) • QB 5(a) • QB 5(b) • QB 5(c) • QB 6(a) • QB 6(b) • QB 6(c) • QB 6(d) • QB 6(e) •
Question B 03 (a) 

Let A be a matrix which transforms and transforms What are the eigenvalues and eigenvectors of A? 
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 an eigenvector is a vector such that, leftmultiplying it by A produces a multiple of itself, i.e., Do you see this property in the provided vectors? 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. From the hint we recall that an eigenvalue, eigenvector pair satisfies, We have in our problem that [1,1] transforms to [2,2], which in matrix form is, Therefore we see that [1,1] is an eigenvector of A with eigenvalue 2. For the other case we have that [1,1] gets transformed to [0,0]. Therefore, and so we see that [1,1] is also an eigenvector with eigenvalue 0. Recall that an matrix has at most n eigenvalues. Therefore since in this case, n=2, we have found all the eigenvalues of the system. We conclude that A has eigenvalues and eigenvectors, 