If A, B & C are matrices & AB=AC then B=C?
as we know it is not always true because when A is singular matrix then B=C not possible
When matrix $A$ is singular and $AB=AC$ then $B=C$ would also be possible. For example, consider A,B,C as zero matrices. You could also prove like as:
Consider square matrices $A,B,C$ are of same size then
$AB=AC \Rightarrow \det(AB) = \det(AC) \Rightarrow \det(A)\det(B) = \det(A) \det(C) $
$\det(A)\det(B) - \det(A) \det(C) =0 $
$\det(A) (\det(B) – \det(C)) = 0$
So, if matrix $A$ is singular then $\det(A) = 0$ which implies either $(\det(B) – \det(C)) = 0$ or $(\det(B) – \det(C)) \neq 0$
when $(\det(B) – \det(C)) = 0 \Rightarrow \det(B) = \det(C)$ then matrix $B$ may or may not be same as matrix $C.$
Singularity or Non- Singularity of matrix A is one reason to say matrix B may or may not be equal to C but based on only singularity of $A$, it would also be suffice to say matrix B may or may not be equal to matrix C.