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

MATH307 December 2005

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Question 03 (a)
Decide whether the following statement is true or false. You need not give a reason. All matrices in this question are square $n\times n$. If $\displaystyle A=A^{-1}$ then every eigenvalue of A is either 1 or -1.

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

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

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 answer is true. We are given that $\displaystyle A=A^{-1}$ which implies that $\displaystyle A^{2}=I$. Let $\displaystyle p(\lambda )$ is the characteristic polynomial of A and $\displaystyle m(\lambda )$ its minimal polynomial. Notice that $\displaystyle m(\lambda )$ divides $\displaystyle q(\lambda )=\lambda ^{2}-1$ (since the matrix itself satisfies its minimal polynomial by the Cayley-Hamilton theorem). Furthermore, p and m also share roots and these form the only roots. Hence all the roots of p must be a subset of the roots of $\displaystyle q(\lambda )$ which are $\displaystyle \pm 1$. These roots are the eigenvalues of the matrix A and so the matrix can only have these eigenvalues.

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. This statement is true. If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A-1 with the same eigenvectors (Av = λv, v = λA-1v, v/λ = A-1v). Hence $\lambda v=Av=A^{-1}v={\frac {1}{\lambda }}v$ or, equivalently (since v ≠ 0), $\displaystyle \lambda ^{2}=1,$ so λ = -1 or λ = 1.

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