In general the determinant of a sum is not the sum of the determinants. In that case I don't know that equality is interesting. In general we can not evaluate a path integral knowing only the endpoints (but conservative vector fields are useful). This arc length integral was fairly easy but in general numerical methods would be required. Those are some of the ways I might use it. So "usually" in some useful sense.
later perusing Math Reviews for the phrase "not true in general" reveals many cases of A implies B but the converse is not true in general Also A implies B however B is not true in general