$A,B,C\in M_n(\mathbb F)$ matrices of the same size such that $rank(A)=r\;\&\;AB=AC.$ Calculate $\max\{\text{rank}(B-C)_{n}\}$
This question is a part of my homework in Linear Algebra, and the title of the homework is "Vector spaces, Linear dependence, Span, and Basis". I have no clue/direction of how to approch this question.
Thank you!