Can a Matrix A² ever equal -I₃ in M₃(ℝ)?

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No matrix A in M₃(ℝ) can satisfy the equation A² = -I₃. The discussion revolves around proving this statement using properties of determinants. It is suggested to compute the determinant of A to aid in the proof. The participants express gratitude for the assistance received in understanding the problem. The conclusion reinforces the impossibility of such a matrix existing.
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Homework Statement


Show that no matrix A ∈ M3 (ℝ) exists so that A2 = -I3

Homework Equations

The Attempt at a Solution


This is from a french textbook of first year linear algebra. I'm quite familiar with properties of matrices but I don't have any idea of how to prove this.

Thanks for the help!
 
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Think about the determinants in the given equation.
 
Compute the determinant of ##A##.
 
I took me a while but I got it. Thanks both of you!
 

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