Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true?

1. A
   If $\det A=\pm 1$, then $A^{-1}$ exists but all its entries are not necessarily integers
2. B
   If $\det A\ne \pm 1$, then $A^{-1}$ exists and all its entries are non-integers
3. C
   If $\det A=\pm 1$, then $A^{-1}$ exists and all its entries are integers
4. D
   If $\det A=\pm 1$, then $A^{-1}$ need not exist