If $A$ and $B$ are both $n \times n$ matrices, and $v$ is a non-zero $n \times 1$ column vector then is it true that if $$ABv = BAv$$ then $$AB=BA$$
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The short answer is NO.
You cannot say $AB = BA$ if $ABv = BAv$ for some vector $v$.
However, if $ABv = BAv$ is true for all vectors $v$ (or) at-least for $n$ linearly independent vectors $v$, then it is true that $AB = BA$.
Suppose $AB = C.$ Then we have $ABv= Cv =BAv$ for all $v.$
Do you feel comfortable saying $BA = C$ now? Do you know any theorems about the uniqueness of matrices of linear transformations under fixed bases?
Edit: In Light of the other answer, i should clarify, this is true only for all $v.$