Must be Non negative.

Taking $m = 1$ and $n = 0$ gives integers $x_1$ and $y_1$ with $ax_1 + by_1 = 1$ and $cx_1 + dy_1 = 0$ . Similarly, taking $m = 0$ and $n = 1$ gives $x_2$ and $y_2$ with $ax_2 + by_2 = 0$ and $cx_2 + dy_2 = 1$ .

Then we compute

$\large\ \left( ad - bc \right) \left( { x }_{ 1 }{ y }_{ 2 } - { x }_{ 2 }{ y }_{ 1 } \right) = (ax_ 1 + by_ 1)(cx_ 2 + dy_ 2) - (cx_ 1 + dy_ 1)(ax_ 2 + by_ 2) = 1\cdot 1 - 0\cdot 0 = 1$ .

Since these are integers, we must have $ad - bc = ±1$ .