Prove or give a counterexample: For $A \in \mathbb{R}^{3\times 3}$, $\det(A + I) = \det(A) + \det(I)$ if $\mbox{tr}(A) = -\mbox{tr}(\mbox{adj}(A))$. Here, $\mbox{adj}(A)$ is the classical adjoint (the transpose of the cofactor matrix).
Thanks in advance!