MathNet 2014 Question 10

Let $A$ be an invertible matrix in $M_4(\mathbb{R})$, such that $\operatorname{tr} A = \operatorname{tr} A^* \neq 0$, where $A^*$ is the adjugate of $A$. Prove that the matrix $A^2 + I_4$ is singular if and only if there exists a nonzero matrix $B$ in $M_4(\mathbb{R})$, so that $AB = -BA$.

Mariean Andronache