Calculate the rank of matrix $B-C$ while $AB=AC$ and $\operatorname{rank}(A) = r$? $A,B,C\in M_n(\mathbb F)$ 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! 
 A: $Im(B-C) \subset null(A)$ then $$\dim Im(B-C)+ \dim Im(A) \leq \dim Im(A) + \dim null (A) = n$$ $$\dim Im(B-C) \leq n -\dim Im(A)$$
A: Hint: Consider a vector $v$ in the column space of $B-C$. Then $v=(B-C)u$ for some vector $u$. Since $AB=AC$, we have $Av=0$. So, $v$ also lies inside the null space of $A$. Now, what is the relationship between the dimension of the column space of $B-C$ and $\mathrm{rank}(B-C)$? What is the dimension of the null space of $A$?
A: Given all the information, let's observe the relation: $AB=AC$ and see if $B\;\&\;C$ can be 'direct complements'(a bit tricky expression if you want a quick and correct explanation from a student like me (: , but play with $\text{symmetric & skew-symmetric}$ matrices) or if $$rank(B)+rank(C)=n\;\&\;rank(B\pm C)=n\implies (B-C)\text{~}D_n\text{~}I_n$$
Although(for $\text{both left & right zero divisors}$): $X,Y\in M_n(\mathbb F),\;X,Y\ne0\;XY=YX=0_n\;\text{is true}$
why not $$rank(A)=r=0?\;\;\text{(there is no constraint on that)}$$
$$A=0_n\implies AX=AY=0\;\forall X,Y\in M_n(\mathbb F)$$
It is excellent if none of the given matrices is regular!
$$A=0_n\implies (B-C)\;\text{can be regular}$$
$$Q.E.D$$
Therefore, the maximum possible range of $(B-C)$ is $n$.
Anyway,if we denote the column or row spaces (in $B\;\&\;C$) with $s_i,s'_i$ or $t_j,t'_j$, then:
$$s_1\oplus s_2\oplus\ldots\oplus s_{n-rank(C)}\oplus s'_{n-rank(C)+1}\oplus\ldots\oplus s'_n=M_n(\mathbb F).$$
In other words, there are $n-rank(X)$ linearly dependent column (row) vectors in $X_n\in(\mathbb F)$. Take that into account if you have to express the solution with respect to an arbitrary $r\in \{0,1,2,\ldots,n\}$
