How many 3x3 and 4x4 matrices with or without complex entries (the distinction is important) square to the negative of the identity matrix?

Question

anonymous · Answer

how many?

beginnersmind · Answer

Deleted previous answer which was probably incorrect :)

anonymous · Answer

I'm compelled to say that only $A=iI$ satisfies the condition $A^2=-I$ for any matrix of size $n$. Perhaps it's possible to prove uniqueness by contradiction, but it's not obvious to me at the moment.

beginnersmind · Answer

Doesn't rotation by pi satisfy in n = 2? I think something similar works in any even dimension.

beginnersmind · Answer

@nettle404

anonymous · Answer

What rotation maps itself to $-I$? My matrix algebra is a bit rusty, so you may have to point out an obvious mistake of mine.

beginnersmind · Answer

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I mean like this.