Here is my thoughts on the question.

Every solution to a linear system can be decomposed into two separate pieces.

where is the complimentary solution (the solution to the homogeneous equation) and is a particular solution.

Since we know the solution is unique for some particular

We have

Since the solution is unique The complementary solution cannot have any free parameters.

e.g the kernel of this Matrix has only one vector in it and since the only solution to the system

is

This will give the desired conclusion.