Question 17676: A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank

Original question and givens

Use the complete ordered integer matrix A=[[1, 0], [0, 1]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).

• G001 — Use the complete ordered integer matrix A=[[1, 0], [0, 1]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-matrix-rank

Step-by-step derivation

rank pivot0 column0 normalized entry0 := div(1, 1)

rank pivot0 column0 normalized entry0 = 1

rank pivot0 column0 normalized entry1 := div(0, 1)

rank pivot0 column0 normalized entry1 = 0

pivot row=0;pivot column=0;nonzero pivot=1;old row=[1, 0]

normalized full row=[1, 0]

rank pivot1 column1 normalized entry0 := div(0, 1)

rank pivot1 column1 normalized entry0 = 0

rank pivot1 column1 normalized entry1 := div(1, 1)

rank pivot1 column1 normalized entry1 = 1

pivot row=1;pivot column=1;nonzero pivot=1;old row=[0, 1]

normalized full row=[0, 1]

complete original square matrix=[[1, 0], [0, 1]];final exact RREF=[[1, 0], [0, 1]];increasing pivot columns=[0, 1]

all rows retained inclzero;rank=2;row operations invertible;row-rank equals column-rank by the finite-dimensional theorem

whole original matrix diagonal product := mul(1, 1)

whole original matrix diagonal product = 1

whole original matrix off-diagonal product := mul(0, 0)

whole original matrix off-diagonal product = 0

whole original matrix determinant difference := sub(1, 0)

whole original matrix determinant difference = 1

original row indices in SUPPLIED order=[0];original column indices in SUPPLIED order=[0];same whole matrix=[[1, 0], [0, 1]]

selected exact square minor=[[1]]

selected supplied-order minor:same1x1 minor=[[1]]

determinant is its sole entry=1

supplied minor size=1;exact signed determinant=1;nonzero=Yes

certificate from THIS minor=rank_at_least_1

Reverse checks

• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute whole original matrix determinant difference with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute whole original matrix off-diagonal product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute whole original matrix diagonal product with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute rank pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute rank pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute rank pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute rank pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals.

Result

([0], [0], [[1]], 1, Yes, rank_at_least_1)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.