Lin. alg. proof: If at most k rows have non-zero entries,...

[buckaroobill]

This was confusing me so any explanations would be appreciated:

Prove the following by using induction on n: Let A, B, and C be n by n matrices such that C = Ax+B. If at most k rows of A have nonzero entries, then the determinant of C is a polynomial in x of degree <= k.