RO-NMO 2014 Problem 114

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