Note that Does that give you any ideas?
You should have learned that det(AB)= det(A)det(B) (as TheEmptySet mentioned). If AB= I then det(A)det(B)= 1 so neither det(A) nor det(B) can be 0.
However, Ackbeet's suggestion gives a more fundamental proof than that. If had an inverse, then there would exist some matrix, B, such that . But that is the same as which says that is inverse to A contradicting the fact that A does not have an inverse.