Linear algebra question?

Question

I have been working on this all day, but I'm still confused as to how to do it and it's REALLY late. If anyone could explain to me how this is done, I would really appreciate it. Thanks in advance!

Let A, B, C, be n by n matrices such that C = Ax + B. Prove that if at most k rows of A have nonzero entries, then the determinant of C is a polynomial in x of degree less than or equal to k.

Answer 1

If at most k rows of A have nonzero entries, then at most k rows of C will have entries with x terms; the remainder will be constants.

Each term in the determinant is a product of one element from each row of C. Since at most k of these can be x terms, the product can have degree at most k. So the determinant is a sum of polynomials of degree k or lower, so is itself a polynomial of degree at most k.