Suppose $ad+bc=n$. It is given that $n$ divides each of $a$, $b$, $c$ and $d$. Let $a=na'$, $b=nb'$, $c=nc'$ and $d=nd'$, where $a', b', c', d'\in\mathbb{Z}$. Then, $$n^2(a'd'+b'c')=n$$$$\implies a'd'+b'c'=\frac{1}{n}$$And as $\frac{1}{n}\in\mathbb{Z}$, $n=\pm 1$, as desired.