Let me see if I understand this correctly
Q: is ab=1
know 1) aba=a which can be rewritten as a(ab-1)=0
know 2) bab=b which can be rewritten as b(ab-1)=0
a or b could equal 0
Therefore C would be insufficient.
Alternative method #2
But plugging in a and b into ab=1, we get
(aba)(bab)=1 which rearranged is (b^3)(a^3)=1
The only way for this formula to be true would be if
b and a were 1
b and a were -1
Method #1 tells us ab=1 which is true, but that a and b could also be 0. Thus inconclusive
Method #2 tells us a and b must both be 1 or both be -1.
Let me know if I'm off here please.