Since b is a 3x1 matrix obtained from Ax, we can easily infer that the dimensions of A and x must be 3xn and nx1. So, if A is a mxn matrix, where m must be equal to 3.
If we have a unique solution for b and, at the same time, no solution for another b, it means that A has linearly independent column vectors, but they are not covering the entire space. So, the number of linearly independent vectors in matrix A must be less than the dimension of b, which is less than 3. So, n should be either 2 or 1. So, the rank can also be either 2 or 1.