I think here we required rank(A | B) and rank(A) to conclude clearly . Becoz;
FOR HOMOGENEOUS SYSTEM :-
1) INCONSISTENT ;- Not possible
2) Unique solution :- Rank == No of variables
3) Many Solution :- Rank < No of variables.
FOR NON - HOMOGENEOUS SYSTEM :-
1) INCONSISTENT ;- when Rank(A|B) !=Rank (A)
2) Unique solution :- when Rank(A|B) =Rank (A) and Rank == No of variables
3) Many Solution :- when Rank(A|B) =Rank (A) and Rank < No of variables.
So It has m linear eqn so maxm rank can be = "m"
and no of variables are = "n"
and to conclude anything Further we need to assume Rank (A|B)=rank(A).
After assuming this only Case (III) can be true.
Becoz m can not be greater than n (Becoz if so , then we can not conclude anything) , hence (II) is false.
When m<n , then all such system has Many solution, Hence (I) is false..