Find conditions on a, b, c, d $\displaystyle \in \mathbb{Z}$ for {(a, b), (c, d)} to be a basis for $\displaystyle \mathbb{Z}\times\mathbb{Z}$. [Hint:Solve x(a, b) + y(c, d) = (e, f) in $\displaystyle \mathbb{R}$, and see when the x and y lie in $\displaystyle \mathbb{Z}$.]

My solution:

x(a, b) + y(c, d) = (e, f)

ax + cy = e

bx + dy = f

x = (ed - fc)/(ad - bc)

y = (af - be)/(ad - bc)

Since x and y $\displaystyle \in \mathbb{Z}$, $\displaystyle ad - bc =\pm 1$ is the required condition.

Am I right?