Why is $\det(e^X)=e^{\operatorname{tr}(X)}$? I've seen on Wikipedia that for a complex matrix $X$, $\det(e^X)=e^{\operatorname{tr}(X)}$.
It is clearly true for a diagonal matrix. What about other matrices ?
The series-based definition of exp is useless here.

If $X$ is upper triangular, then this is clear.
If the claim holds for a matrix $Y$, then it holds for any $X$ similar to $Y$.
By Jordan decomposition, each $X$ is similar to an upper triangular matrix $Y$ (of a special form, but never mind).
Thus, the claim holds for all $X$.
