Thanks for the response. The motivation for the question came from a totally different angle (optics); I didn't think of the problem in terms of factoring. If I follow your answer correctly this would imply that adding the requirement that the matrix is invertable would mean that only the case $n=2$ has solutions. This negative result would have value on its own. Do you have a good reference for the irreducibality of the fermat curve? also the fermat curve is in only 3 variables, can I assume that the general $\sum_1^n x_i^n$ is also irreducible?