Given a linear system Ax = b, if we know x and b, could we retrieve the original A?

By given a linear system Ax = b, more often, we think about how to solve x if we have known A and b. Now, let us rethink this problem if we have known x and b, could we retrieve the original A? Is the retrieved A unique?

Without generosity, we consider x is an m×1 vector, b is an n×1 vector, and A is an n×m matrix. To vividly present the solution, let us try a toy example as follows.

Here, matrix A could also be treated as a linear transformation that maps x to b…