Assume that for a non-homogenous system of m linear equations in n unknowns, the rank of the coefficient matrix is less than the rank of the augmented matrix. What conclusion can necessarily be drawn?
(A) The system is inconsistent (no solution exists) (B) The system has infinitely many solutions (C) The system has a unique solution (D) The system has a finite number of solutions but more than one (E) Additional information is needed before a conclusion can be reached
The answer is A but I'm pretty lost when it comes to this. Any insight is much appreciated.
The assumption implies that the augmented matrix has at least one additional pivot than the original matrix when row-reduced. This corresponds to a column of zeros with a nonzero entry in the augmented part. This implies $0=1$, so that the system is inconsistent.