I understand the proof of the identity in the title for $A$ Hermitian. One uses that any Hermitian matrix can be diagonalized as $A = X \Lambda X^{-1}$, such that
$$
\det{A} = \prod_i \lambda_i,
$$
and we have
$$
\exp(Tr(\log(A)) = \exp(Tr(X\log\Lambda X^ {-1}) = \exp(\sum_i\log(\lambda_i)) = \prod_i \lambda_i.
$$
However, is it possible to show the identity for $A$ not Hermitian? My motivation for this question is that in physics the identity is often used without it being clear that $A$ is Hermitian.
Best Answer
If $\ln(A) = B$, the identity says
$$ \det(\exp(B)) = \exp(\text{Tr}(B)) $$
which is more usual form for this identity, true for all $n \times n$ matrices $B$ over $\mathbb C$ (avoiding questions about whether $\ln(A)$ is defined, and which of the possible logarithms to use).
One way to do this is to show it first for diagonalizable matrices $B$, then use the fact that diagonalizable matrices are dense and both sides of the equation are continuous functions of $B$.
A second way is to use Jordan canonical form.
A third way is to note that both $\det(\exp(tB))$ and $\exp(t \text{Tr}(B))$ satisfy the differential equation $y' = \text{Tr}(B) y$ with initial value $y(0) = 1$.