You should know by now, besides Cramer's Rule, also the basic formula , where the
classical adjoint of the matrix (whose entries are the minors of A ), the determinant of A,
and the unit matrix.
From this it follows that if A is invertible, then , and then it is straightforward that if A is an
integer matrix and , then also is an integer matrix.
The other direction follows from Cramer's rule: in , the entry ij of ,which of course
in an integer, is given by , where the ij-entry of . Continue from here.