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

Given that all the elements of matrix A are integers.
So, the cofactor of each element is also an integer.
So, the adjoint A has all its elements as integers.
Now, also given $detA=\pm 1$
So, $A^{-1}$ exists.
We have $A^{-1}=\frac{adjA}{|A|}$
$A^{-1}=\frac{adjA}{\pm 1}$
So, all the elements of $A^{-1}$ are integers.