Science:Math Exam Resources/Courses/MATH221/April 2009/Question 12 (a)

MATH221 April 2009

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Question 12 (a)
Mark each statement either True or False. You do not have to justify your answer. If A is a real $\displaystyle n\times n$ matrix with $\displaystyle A^{2}=-I$, then n is even.

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
What does the Cayley-Hamilton Theorem tell us in this case?

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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 answer is true. The Cayley Hamilton theorem tells us that the eigenvalues are roots of $\displaystyle q(\lambda )=\lambda ^{2}+1=0$. Thus, exactly one of $\displaystyle \lambda +i$ or $\displaystyle \lambda -i$ contains an odd number of factors in the matrix $\displaystyle p(\lambda ){\text{det}}(A-\lambda I)$. This means the constant term of $\displaystyle p(\lambda )$ (which is the product of the roots) must be strictly non-real, a contradiction since the matrix was assumed to be real.

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