Why is AES not a Feistel cipher? I am studying for an exam right now.  And I wanted to make sure I got this point correct.

AES is not a Feistel cipher because the operations in AES are not invertible.

Is the above statement correct? If not, why isn't it a Feistel cipher?
Well, AES is not a Feistel cipher because it's a substitution-permutation network instead. If I were taking a test that asked me why AES was not a Feistel cipher, this would be my argument: namely, that the structure of substitution-permutation networks is fundamentally different from that of Feistel networks. (Here one could elaborate on invertibility and other differences.)

That said, your statement is not correct. In a Feistel cipher, the round function is not necessarily invertible (DES's round function is not), but in AES, like any substitution-permutation network, the rounds are invertible. This is a property of the construction itself.