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

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Homework Help Overview

The discussion revolves around the question of whether a matrix \( A \) in \( M_3(\mathbb{R}) \) can satisfy the equation \( A^2 = -I_3 \). This is a topic within linear algebra, specifically concerning properties of matrices and determinants.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of the equation by considering the determinants of the matrices involved. There is an emphasis on understanding the properties of the determinant in relation to the equation.

Discussion Status

The discussion has seen contributions that suggest examining the determinants, with at least one participant indicating they have reached an understanding after engaging with the responses. However, there is no explicit consensus on the final outcome of the inquiry.

Contextual Notes

The original poster notes familiarity with matrix properties but expresses uncertainty about how to prove the statement, indicating a potential gap in understanding that is being explored through the discussion.

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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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