Computing ABx given Ax

Question

In general matrix multiplication does not commute, but is there any information about one ordering of the product contained in a different ordering (assuming the dimensions match)? For example, if I know $y=Ax$, I can easily get $BAx$ from $B$ and $y$ as $BAx=By$. But, is there any way to compute $ABx$ from just $B$ and $y$? If not, are there any conditions under which it's possible (excepting the obvious $AB=BA$)?

Answer 1

Let $$A=\begin{pmatrix} 0 & 1\\ 0&0\end{pmatrix}\qquad B=\begin{pmatrix} 0 & 1\\ 1&0\end{pmatrix} \qquad x=\begin{bmatrix} \lambda \\ 0\end{bmatrix} $$ Then $ABx=x,$ while $y=Ax=0.$ Thus $ABx$ cannot be determined basing on $y$ and $B.$