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

Put content here**Theorem: Echelon Form and Consistency**
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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.
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**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`.
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**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.
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**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
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**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.

# Parents

* Linear systems and echelon matrices