I guess daniel-levin meant to say "invertible matrix with integer coefficients whose inverse also has invertible coefficients". The fact isn't too hard to see:

* If M has integer coefficients, then det(M) is an integer. * det(inv(M)) = 1/det(M) * Since M has integer coefficients, det(M) is an integer * Since inv(M) has integer coefficients, det(inv(M)) is an integer * So det(M) and 1/det(M) are both integers, so det(M) is either 1 or -1