Does putting a different quantifier changes the veracity of this linear algebra exercise? In this exercise, I'm asked to indicate - and justify - whether the following is true or not.
Let $A,B,C$ are matrices with real entries, with suitable dimensions. 
If $AC=BC$, then $A=B$.
My question is whether I should interpret this statement as $\exists C$ or $\forall C$.
In the case of $\exists C$, we can find $D\neq 0$, such that $DC=0$, making it possible to have $A\neq B$.
In the case of $\forall C$, then when we have $C=I$, we must have $A=B$. (edit: Completely wrong understanding of$\forall$ quantifier) 
Am I right in this assessment?
Any help would be appreciated.

"Let $A,B,C$..." means that $A,B,C$ may be any triple of matrices. 
Thus you're asked to prove that
$$\forall A,\forall B, \forall C, (AC=BC\Rightarrow A=B)$$
This is clearly false (take $C=0$). 
