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Characterization of some singular matrices of the form I+A²

Source: Romania National Olympiad 2014, Grade XI, Problem 4

March 3, 2019
linear algebramatrix

Problem Statement

Let A∈M4(R) A\in\mathcal{M}_4\left(\mathbb{R}\right) be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that A2+I A^2+I is singular if and only if there exists a nonzero matrix in M4(R) \mathcal{M}_4\left( \mathbb{R} \right) that anti-commutes with it.
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