On page 229 of Cracking the GRE Math Subject test, it's written:

"Notice that if we replace R by Z in GL(n,R), we get a monoid but not a group, because the inverse of a nonsingular matrix with integer entries may not have integer entries. However, if the determinant of such a matrix is equal to 1, then the entries of the inverse will also be integers..."

Why is it the case that if the determinant of such a matrix is 1, then the entries of the inverse will also be integers?