If $Ax = Bx$ for all $x \in C^{n}$, then $A = B$.
For the sake of an answering: Yes, your proof is correct.
If $Ax = Bx$ for all $x \in C^{n}$, then $A = B$.
For the sake of an answering: Yes, your proof is correct.
For the sake of an answering: Yes, your proof is correct.