The echelon form can be used to determine if a linear system is consistent.

Theorem: Echelon Form and Consistency

A linear system is consistent if and only if the rightmost column of its augmented matrix is NOT a pivot column in the echelon form.

Statement: Given an augmented matrix [A | b], the system Ax = b is consistent if and only if the echelon form of [A | b] has no row of the form:

[ 0  0  ...  0  |  c ]

where c ≠ 0.

Explanation: Such a row corresponds to the equation 0 = c (with c ≠ 0), which is a contradiction. This happens precisely when the last column (the constants) becomes a pivot column during row reduction.

Procedure to check consistency:

1. Row reduce the augmented matrix to echelon form
2. Examine each row: if any row has all zeros in the coefficient part but a nonzero entry in the constant column, the system is inconsistent
3. Otherwise, the system is consistent

Corollary: A homogeneous system Ax = 0 is always consistent (the trivial solution x = 0 always exists), since the last column is always zero and can never become a pivot.