$Matrices\:A\:and\:B\:are\:row\:equivalent\:if\:and\:only\:if\:\left(iff\right)\:they\:can\:be\:reduced\:to\:the\:same\:row\:echelon\:form.$
This result was introduced in my class but I don't have it's proof, I have developed a thought process for one side of the proof, if matrices $A$ and $B$ have the same RREF then they are row equivalent.
Elementary row operations can be thought of as multiplication of matrices with elementary matrices. $$RREF\:=\:E_1E_2E_3...E_nA$$ $$RREF\:=F_1F_2.....F_nB$$
Multiply both sides of the second equation by the inverse of the elementary matrices after replacing RREF as mentioned in the first equation. I can do the same for the other side of the proof too right? $$A\:=\:E_1E_2.....E_nB$$ $$RREF\left(B\right)\:=\:F_1F_2....F_nB$$ $$B\:=\:F_1^{-1}F_2^{-1}......F_n^{-1}RREF\left(B\right)$$ $$A\:=\:E_1E_2.....E_n\left(\:F_1^{-1}F_2^{-1}......F_n^{-1}RREF\left(B\right)\right)$$ I wanted to verify whether this proof is accurate, I am also open to other proofs probably more involved with the fundamental since I am just starting off with Linear Algebra.