Given nxn matrices A and B, and an n-vector x, are there any conditions that can guarantee Ax=Bx implies A=B? I started thinking about this well working on an assignment. It is clearly not always true, since you can easily think up 2x2 examples where it is not. You can also think up examples where the determinants are non-zero, and not equal to each other, where it still doesnt hold. What conditions if any would make this implication true? Thank you.