Building positions from three baskets

A desk can trade three stock baskets, long or short, in any size. One unit of basket 1 holds 1 share of X and 2 of Y. One unit of basket 2 holds 1 share of Y and 3 of Z. One unit of basket 3 holds 2 of X, 5 of Y and 3 of Z. Writing each basket's holdings in (X, Y, Z) as a row:

Compute $\det W$. Can the desk end up holding exactly one share of Z and nothing else, using only the baskets? Which target holdings $(x, y, z)$ can it reach?