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?

## SL(n, Z)

### Re: SL(n, Z)

A * adj(A) = adj(A) * A = det(A) * I

If A only contains integers, then the adjugate matrix also only contains integers (arises from entries of A without division). So we see A^(-1) exists in GL(n,Z), in which case it equals adj(A), if and only if det(A) = 1.

If A only contains integers, then the adjugate matrix also only contains integers (arises from entries of A without division). So we see A^(-1) exists in GL(n,Z), in which case it equals adj(A), if and only if det(A) = 1.