Matrix rank

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

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 — The finite matrix is complete.
• Linear combinations and span — Rows and columns generate their respective spaces.
• Dimension of a finite-dimensional vector space — 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.