Suppose Ax = b and Cx = b have same solutions for every b. Is it true that A = C?

Answer 1

Either the professor did not include enough information or you are just suppose to prove this by contradiction

Proof

Let A not be equal to C and let x = 0 then Ax=0=b and Cx=0=b.

Hence, when x=0 A and C can be anything.

Answer is FALSE

Q.E.D.

If you teacher said x does not equal the zero vector in the question, then that is a different story.