Suppose $A$ is a matrix in $\mathbb{R}^{k \times k}$. Show that $$\det e^A = e^{\text{Tr } A},$$ where $\text{Tr } A$ is the trace of the matrix $A$.
This is marked as a "starred exercise" in Analysis in Euclidean Spaces by Ken Hoffman, and I have no idea how to even get started on this question.
It's straightforward to verify this for diagonal matrices over the complexes.
Consequently, it's easy to show for diagonalizable matrices over the complexes.
All of the functions involved are continuous; since the diagonalizable matrices are dense in the space of all matrices, the identity holds for all matrices over the complexes.
Finally, every real matrix is a complex matrix.