Math 2.0

Linear and multilinear algebra; matrix theory Proposition

A pivot in every row of the coefficient matrix implies consistency

Builds on Coefficient matrixPivotConsistent and inconsistent systemsSystem of linear equations

Statement

Take a system of linear equations and its coefficient matrix.

Suppose that each row of the coefficient matrix contains a pivot position.

Then the system is consistent, 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.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Coefficient matrix, concept, no evidence levelCoefficient matrixconceptLinear equation, definition, no evidence levelLinear equationdefinitionPivot, concept, no evidence levelPivotconceptSystem of linear equations, concept, no evidence levelSystem of linear equa…conceptSystem of linear equations, definition, no evidence levelSystem of linear equa…definitionConsistent and inconsistent systems, definition, no evidence levelConsistent and incons…definitionA pivot in every row of the coefficient matrix implies consistency, statement, no evidence levelA pivot in every row …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Coefficient matrix concept
  2. Linear equation definition
  3. Pivot concept
  4. System of linear equations concept
  5. System of linear equations definition
  6. Consistent and inconsistent systems definition
  7. A pivot in every row of the coefficient matrix implies consistency statement

Further reading

  1. Web https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section1-4.xml#L633, src/section1-4.xml:633
About this page
TagHJTF7B36W9
URIm2://HJTF7B36W9
Kindstatement, proposition
WritingW0 stub
MSC15A06
StatusThis copy of the site has no build data. Run m2 status HJTF7B36W9.
AboutCoefficient matrix
UsesCoefficient matrix, Pivot, Consistent and inconsistent systems, System of linear equations
Changed4 November 2026

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