---
tag: "HJTF7B36W9"
kind: statement
name: A pivot in every row of the coefficient matrix implies consistency
role: proposition
msc:
  - 15A06
sources:
  - url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section1-4.xml#L633
    locator: src/section1-4.xml:633
uses:
  - 0HQFZAP3ND
  - 0J0BBCMJ5W
  - 04JAAVHDDM
  - 0GT4WWGWH2
about:
  - 0HQFZAP3ND
---

## Statement

Take a [[system of linear equations|0GT4WWGWH2]] and its [[coefficient matrix|0HQFZAP3ND]].

Suppose that each row of the coefficient matrix contains a [[pivot|0J0BBCMJ5W]] position.

Then the system is [[consistent|04JAAVHDDM]], so it has at least one solution.

## Remarks

The condition concerns only the coefficient matrix, not the augmented matrix.

The converse fails: a consistent system can have a row of the coefficient matrix with no pivot.