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

[AllRelative]

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!

 

[andrewkirk]

Think about the determinants in the given equation.

 

[fzero]

Compute the determinant of $A$.

 

[AllRelative]

I took me a while but I got it. Thanks both of you!