TITLE: Find all $n\times n$ matrices $A$ satisfying $\det(I+A^n)=(1+\det(A))^n$
QUESTION [8 upvotes]: Problem: Find all $n\times n$ matrices $A$ satisfying
  $$\det(I+A^n)=(1+\det(A))^n.$$

Clearly, the identity matrix $I$ works because
$$\det(I+I^n)=\det(2I)=2^n=(1+\det(I))^n.$$
Are there any other solutions?

REPLY [5 votes]: We just need that:
$$\prod_{\sigma\in\text{Spec}(A)}(1+\sigma^n)=\left(1+\prod_{\sigma\in\text{Spec}(A)}\sigma\right)^n \tag{1}$$
that can be achieved in many ways. However, if $\text{Spec}(A)\subset[0,+\infty)$ (that happens if $A$ is positive semi-definite, for instance), by the super-additivity of the geometric mean (LHS$>$RHS) it follows that $(1)$  may hold only if $\text{Spec}(A)=\{\sigma\}$.