I have 2 questions.
(1) Assume problem B is NP-complete and it is polynomial time reducible to problem A. Also the problem A is polynomial time reducible to problem B. Then is A NP-complete?
We already know that A is NP-hard because B, which is NP-complete, can be reduced to A in polynomial time, but is it possible to know that A is NP-complete? I think I should use the information that $A\leq_PB$ but, I don't know how to use it.
(2) Assume that A is an NP-complete problem. If the problem B can be reduced into A in polynomial time, does this give any information about B?
I think this is related to question (1). I believe that B is NP-hard, but I'm wondering how to prove it mathematically.