So your first question is asking whether there exist scalars such that
This equation you can translate into a normal matrix problem of the form . Row reduce that problem. If you find any solutions (one or infinitely many), then the answer is yes. If you find that there are no solutions, then the answer is no.
Your second question, again, is asking whether there exist scalars such that for your arbitrary This, again, can be translated into a standard matrix problem. Since this problem will result in a square matrix, you can use the theory of determinants to help you out.