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Thread: If A^2 = -I then prove...

  1. #1
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    If A^2 = -I then prove...

    Given an n x n matrix A with real entries such that A^2 = -I, prove the following about A:

    1) n is even
    2) A has no real eigenvalues
    3) det A = 1
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hints :

    1) $\displaystyle (\det A)^2=(-1)^n$

    2) If $\displaystyle \lambda \in\mathbb{R}$ is an eigenvalue of $\displaystyle A$ , then $\displaystyle Ax=\lambda x$ for some $\displaystyle 0\neq x\in \mathbb{R}^n$ .

    Prove that $\displaystyle (\lambda^2+1)x=0$ (contradiction).

    3) No hint, try it.


    Fernando Revilla
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