[Math] Relationship between the trace of a matrix and the trace of its adjoint

Question

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!

Answer 1

Hint: evaluate the characteristic polynomial of $A$ (that is $\det(\lambda I - A)$) at $\lambda = -1$. What are the coefficients to the various powers of $\lambda$?