Find Real nxn Matrices with Negative Identity Power | Matrix Power Question

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Homework Statement


For what positive integers x and n is there a real nxn matrix whose d-th power is -I (negative identity matrix)


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The Attempt at a Solution



I don't really understand this problem, aren't there endless possibilities? Also, why does it matter what size the matrix is? :confused:
 
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What's this mysterious integer x? Is it the same as d?
 
Yeah sorry that's supposed to be a d. I'm assuming there is a solution to this other than the negative identity matrix?
 
Well, yes, e.g. A=[[0,1],[-1,0]] is a 2x2 matrix and A^2=-I.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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