This is an exercise question from the first chapter of Linear Algebra by Hoffman and Kunze. But it seems to be quite difficult to solve even with knowledge of determinants and other relevant topics. I am curious to know if it is even possible to solve with the knowledge of the material from (a) the first chapter or even (b) an entire course on linear algebra.
The point of this question is not only to come up with a proof. But a proof that can be arrived at with limited prior knowledge.