In a system of equations AX = B. by Cramer's rule if det(A)=0, then the system will have no solution but on the other hand, in non-homogeneous system of equations suppose the rank of square matrix nxn is (n-1) and rank of the augmented matrix[A|B] is also having the same rank but then we say that system is consistent, having rank < no. of variables implies infinitely many solutions.
how is this correct? I mean if the rank of A is n-1 < n and rank(A) = rank(A|B), then it also implies that determinant of square matrix nxn is zero but then we say it is consistent.
pls correct me if my theory is wrong too.