Science:Math Exam Resources/Courses/MATH307/December 2005/Question 03 (f)

MATH307 December 2005

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Question 03 (f)
Decide whether each of the following statements is true or false. You need not give a reason. All matrices in this question are square $\displaystyle n\times n$. If 2 is an eigenvalue of $\displaystyle A$ then 6 is an eigenvalue of $\displaystyle A+A^{2}$

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
Start of with the associated eigenvector $\displaystyle v$ for the eigenvalue 2. Try applying the matrix $\displaystyle A+A^{2}$ to this vector.

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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. The statement is true. Let $\displaystyle v$ be the eigenvector of $\displaystyle A$ associated to the eigenvalue 2. Then $\displaystyle (A+A^{2})v=Av+A^{2}v=2v+A(Av)=2v+A(2v)=2v+2Av=2v+4v=6v$ and so 6 is an eigenvalue for $\displaystyle A+A^{2}$.

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