Question

Let \(A\) be a square matrix with integer elements. For each of the following give a proof or counterexample.
a) If \(\operatorname{det}(A)=\pm 1\), then for any vector \(y\) with integer elements there is a vector \(x\) with integer elements that solves \(A x=y\).
b) If \(\operatorname{det}(A)=2\), then for any vector \(y\) with even integer elements there is a vector \(x\) with integer elements that solves \(A x=y\).

c) If all of the elements of \(A\) are positive integers and \(\operatorname{det}(A)=+1\), then given any vector \(y\) with non-negative integer elements there is a vector \(x\) with non-negative integer elements that solves \(A x=y\).
d) If the elements of \(A\) are rational numbers and \(\operatorname{det}(A) \neq 0\), then for any vector \(y\) with rational elements there is a vector \(x\) with rational elements that solves \(A x=y\).