matrices with a fun property

let S be a unit square of 2d vectors (the set of all vectors with x and y between 0 and 1 yada yada) and A some 2x2 matrix

and imagine a function f that applies A to a vector in S, and then takes its new coordinates mod 1

so if , for some vector v , Av is (2.75, 1.5) , then f(v) is (0.75, 0.5)

of course this function f maps S to S :3

now , curiously , for some choices of A this function is bijective! (i believe thats the correct word for it atleast🤭)

an example is [ [2 1] [1 1] ] or [ [1 0] [N 1] ] for whatever N

i cant seem to figure out the pattern of which sorts of numbers work , tho o . o