Example problems sturm louiville

Is there an example of a 3×4 matrix A and a 3×1 vector b⃗ such that the system with augmented matrix [A|b⃗ ] has a unique solution. If so, give an example. If not, why not?

10. Let A be an m × n matrix. Prove or give a counterexample: If Ax = 0 has only the trivial solution x = 0, then Ax = b always has a unique solution.

Solve the system. If there's no unique solution, label the system as either dependent or incomsistent.

2x+y+3z=12

x-y+4z=5

-4x+4y-4z=-20

a. Dependent system

b. inconsistent system

c.(1,4,2)

d. (4,1,2)