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Math Help - Squares of nxn matrices

  1. #1
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    Squares of nxn matrices

    The problem:

    Let A be an nxn matrix. Is it possible for A^2 + I = 0 in the case where n is odd? What about when n is even? (I is the identity matrix)
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    Squaring any nxn matrix to achieve (-1) down the main diagonal is impossible, right? Certainly this can't be the answer that they're looking for though. I have a hunch that there's a more eloquent way to say this using determinants, but I don't know for sure... any suggestions?

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by n3mo View Post
    The problem:

    Let A be an nxn matrix. Is it possible for A^2 + I = 0 in the case where n is odd? What about when n is even? (I is the identity matrix)
    ------
    Squaring any nxn matrix to achieve (-1) down the main diagonal is impossible, right? Certainly this can't be the answer that they're looking for though. I have a hunch that there's a more eloquent way to say this using determinants, but I don't know for sure... any suggestions?

    Thanks in advance!
    If A^2 + I = 0 \implies A^2 = - I \implies [\det (A) ]^2 = \det (-I) = (-1)^n
    And if n is odd then RHS is negative and LHS are is non-negative.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    If A^2 + I = 0 \implies A^2 = - I \implies [\det (A) ]^2 = \det (-I) = (-1)^n
    And if n is odd then RHS is negative and LHS are is non-negative.
    I follow you until you get to =(-1)^n. With either a 3x3 or 4x4 matrix you get a determinant of 1 after squaring them. In both cases the determinants alone become 1 + 1 = 0 for the original equation. Thus it fails in either case-- odd or even. Can you explain what I'm missing in regards to the =(-1)^n part?

    Thanks for the help by the way!
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  4. #4
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    Quote Originally Posted by n3mo View Post
    Let A be an nxn matrix. Is it possible for A^2 + I = 0 in the case where n is odd? What about when n is even?
    If A = \begin{bmatrix}\phantom{-}0&1\\-1&0\end{bmatrix} then A^2=-I. A similar construction works for any even value of n.

    When n is odd it can't be done (with real scalars), as explained by ThePerfectHacker.
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