In the name of Allah, Most Compassionate, Most Merciful,
A female is encouraged to remain within the confines of her home as much as possible. She should not come of the home without need and necessity.
Allah Most High says:
?And stay in your houses and do not adorn yourselves like in the times of ignorance.? (Surah al-Ahzab, 33)
However, if there is a need, then it will be permissible for her to emerge outside, provided the rules of Hijab are fully observed.
As far as the driving of cars is concerned, it would be permissible, subject to the following conditions:
a) There is a genuine need. For example, the husband can not drive due to some reason or there is an emergency, etc... This shows that at times when the husband is being chauffer-driven by his wife, it will not be permissible.
b) There is no alternative.
c) There is no apparent fear of life or respect. If there is just the possibility, then it would be permissible. The argument that women will have to come into contact with strangers in case of an accident, etc is not valid, as coming into contact with strangers can also occur whilst she is walking. When a woman goes to purchase something, she may no doubt have to converse with non-Mahram men.
d) The rules of Hijab are fully observed.
e) If she is driving without the company of any of her Mahrams, then it should only be within 48 miles.
In conclusion, it is permitted for a woman to drive a car with adherence to the aforementioned conditions. However, to avoid it is always advisable; rather staying within the home (as much as possible) and not emerging outside should be practice.
And Allah Knows Best Muhammad ibn Adam