In numerical analysis lingo, you are more or less asking if matrix multiplication is [backward stable](https://en.wikipedia.org/wiki/Numerical_stability). The answer seems to be no: see [Section 3.5](https://books.google.it/books?id=7J52J4GrsJkC&lpg=PR2&dq=accuracy%20and%20stability%20of%20numerical%20algorithms&pg=PA69#v=onepage&q&f=false) of Higham, *Accuracy and stability of numerical algorithms*. I am unable to locate an explicit counterexample quickly, though.