# Matrix rank

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Matrix rank is the dimension of the row or column space and can be computed as the number of pivots after exact rank-preserving elimination.

mko-matrix-rank · definition · v0.10.0



## Mathematical statement

This slice handles 2-by-2 or 3-by-3 integer matrices. A nonzero r-minor gives only a lower bound. An upper bound requires all larger minors to vanish or a complete exact RREF. A zero determinant excludes full rank without determining the exact lower rank; equality of row rank and column rank is a theorem.





## Representative formula (exact TeX)

R(A)





## Assumptions


- The complete matrix is explicit and Gaussian elimination uses exact rational arithmetic.
- Every row operation is explicitly rank-preserving.





## Symbol scope


- R — matrix rank
- A — specified matrix





## Intuition, definition, and misconceptions

The pivot count in the complete RREF gives rank.

Minor certificates distinguish lower and upper bounds.

A zero determinant does not mean rank zero; equality of row and column rank is a theorem obligation.

Do not claim exact rank from a zero determinant or a few sampled row operations.





## Adopted derivation guide

[Full RREF and rank bounds](https://ocme.evemisslab.com/en/derivations/derivation-mko-matrix-rank-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/mko/mko-matrix/) — The finite matrix is complete.
- [Linear combinations and span](https://ocme.evemisslab.com/en/mko/mko-linear-combination-span/) — Rows and columns generate their respective spaces.
- [Dimension of a finite-dimensional vector space](https://ocme.evemisslab.com/en/mko/mko-vector-space-dimension/) — Rank is the dimension of the relevant span.





## Verification boundary

Only exact 2-by-2 and 3-by-3 matrices are handled.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add rank-nullity and matrix decompositions.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/matrix-rank.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-matrix-rank/index.md)