I have a simple question which is troubling me.
I have seen this theorem in Linear Algebra which I quote here :
The Row Echelon Form of an Inconsistent System: An augmented matrix corresponds to an inconsistent system of equations if and only if the last column (i.e., the augmented column) is a pivot column.
Suppose I have a RREF augmented matrix with the last row containing $0s$ in the coefficient and 1 in the augmented column ( I mean $0 0 ... 0|1$ as the last row). Surely, the presence of this row would make the whole system inconsistent, as it would imply that $0=1$, which is not possible.
But, what I don't understand is that for this to happen, why is there a need for all the other entries of the last column to be zero. This doubt stems from the definition of pivot column which says it is a column containing pivot, and hence we know that all the entries in a column containing pivot must be zero.
But I don't see why other entries should be zero. In all the examples of inconsistent system RREF matrices also the last column is always having all the entries except the pivot to be zero.
I hope I made myself clear. I would be grateful to you if you can help me with this.