Provide an example of an augmented matrix in reduced row echelon form that represents an inconsistent system of equations and lacks a pivot in at least one column.

An augmented matrix in reduced row echelon form that represents an inconsistent system of equations and lacks a pivot in at least one column could look like the following:

``` ⎡ 1 0 | a ⎤ ⎢ 0 1 | b ⎥ ⎣ 0 0 | c ⎦ ```

Where `a` and `b` are any numbers and `c` is a non-zero number. The absence of a pivot in the third column ( the coefficient columns) is not a problem for consistency, but the fact that there is a non-zero entry in the last row, in the column augmented (the constants from the right-hand side of the equations), without a corresponding pivot in the coefficient columns indicates that the system is inconsistent. This is because the last row corresponds to the equation `0x + 0y = c`, which cannot be true if `c` is any non-zero number. If `c` were zero, the system would not be inconsistent, but since `c` is not zero, this means that there is no solution to the system of equations.