The system Ax=bAx = bAx=b is inconsistent if:
rank(A)=rank(A∣b)rank(A) = rank(A|b)rank(A)=rank(A∣b)
rank(A)<rank(A∣b)rank(A) < rank(A|b)rank(A)<rank(A∣b)
rank(A)>rank(A∣b)rank(A) > rank(A|b)rank(A)>rank(A∣b)
The matrix AAA is square.