If X is the necessary condition of A, then it doesn't follow that X is sufficient. However, if X were a sufficient condition, would it also follow that X is a necessary condition? Put otherwise, is a sufficient condition simply a set of all necessary conditions that conjointly would guarantee that A obtains?
Being able to drive is necessary, but is not sufficient, for qualifying as a good driver.
Being able to drive well is both a necessary and sufficient condition to qualify as a good driver.
The short answer is "no". Your main question is whether (2) follows from (1);
(1) P → Q [ P is a sufficient condition for Q ],
(2) Q → P [ P is a necessary condition for Q ].
The reason the entailment from (1) to (2) doesn't hold is that it's possible that Q follow from some proposition R that is not equivalent to P. The only instance where the entailment is realized is one where all necessary conditions for Q are logically equivalent to P.
A note on your examples. It's true that you must drive to be a good driver: GD → D. And it's also true, since it's a hidden tautology, that you must drive well to be a good driver (and vice versa): GD ↔ DW. But since not all drivers are good drivers, the entailment doesn't hold.
To be an elephant is sufficient but not necessary for being a mammal.
To be a mammal is necessary but not sufficient for being an elephant.
To qualify as a good driver, it is sufficient to be able to drive well and be female. However, it is not necessary, because also men can be good drivers.
Consider the set W of all objects and the set X of all desired objects, giving U=W\X. A sufficient condition is a predicate s(w) such that s(w) is false for all elements in U. A necessary predicate is a predicate n(w) such that n(w) is true for all elements in X.
Just one quick example:
Two numbers are coprime if they do not share any common factors (other than 1) and are not equal.
From the definition, it is sufficient to show that two numbers are prime numbers to show that they are coprime, but it isn't necessary: 8 and 9 are coprime, but neither are prime.
Thus, not all sufficient conditions are necessary.
Per this answer, 'A is sufficient for B.` means that 'A is a subset of B'.
A picture and real-life example may aid to understand the following:
Your main question is whether (2) follows from (1);
(1) P → Q [ P is a sufficient condition for Q ],
(2) Q → P [ P is a necessary condition for Q ].
The reason the entailment from (1) to (2) doesn't hold is that it's possible that Q follow from some proposition R that is not equivalent to P. The only instance where the entailment is realized is one where all necessary conditions for Q are logically equivalent to P.
The above is exemplified in the picture below, if P = Northern Ireland, Q = UK, R = Great Britain.
Then being in Northern Ireland is sufficient for being in the UK,
but is NOT necessary for being in the UK, because one can also be in U.K. by being in
Great Britain.
