Let $A(t)$ be a smooth function from $[-1,1]$ to the $n \times n$ complex matrices. Define the time ordered exponential $$\prod_{-1}^1 \exp(A(t) dt)$$ as in this question, as the limit of Riemann products $\prod_{i=1}^n \exp(f(t^{\ast}_i) \ \delta t_i)$.

The actual quantity I am interested in is $$B(r) = \prod_{-1}^1 \exp(r A(t) dt)$$ as $r \to \infty$.

As $r \to 0$, there is a known power series expansion for $B(r)$ called the Magnus series. As $r \to \infty$, I would expect there to be something like the stationary phase approximation, but I haven't been able to find it or figure it out.

I should mention that in my situation, $A(t)$ obeys $$A(-t) = A(t)^{\ast} \quad (\dagger)$$ where $\ast$ is conjugate transpose. Condition $(\dagger)$ implies that $B(r)$ is Hermitian. I don't know whether this is helpful in any way.