The echelon form can be used to determine if a linear system is consistent.
Created almost 9 years ago, updated 5 months ago
ID23771
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:
Row reduce the augmented matrix to echelon form
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
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.
**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.