# Question 17691: A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank [Return to catalogue page 89](https://ocme.evemisslab.com/en/questions/page/89/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r01-matrix-rank-foundations/#q-20261014-advance-20261103-r01-matrix-rank-0041) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0034-17691 · q-20261014-advance-20261103-r01-matrix-rank-0041 ## Original question and givens Use the complete ordered integer matrix A=[[1, 15], [0, 16]], 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, 15], [0, 16]], 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](https://ocme.evemisslab.com/en/mko/mko-matrix-rank/) ## Step-by-step derivation S001 · calculationBefore rank pivot0 column0 normalized entry0 := div(1, 1) After rank pivot0 column0 normalized entry0 = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore rank pivot0 column0 normalized entry1 := div(15, 1) After rank pivot0 column0 normalized entry1 = 15 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore pivot row=0;pivot column=0;nonzero pivot=1;old row=[1, 15] After normalized full row=[1, 15] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rationals.The inverse operation multiplies back;do not approximate by floats. Step identifiers and dependenciesRule: rank_nonzero_pivot_entire_row_scaling · Uses: G001, S002 S004 · calculationBefore rank pivot1 column1 normalized entry0 := div(0, 16) After rank pivot1 column1 normalized entry0 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore rank pivot1 column1 normalized entry1 := div(16, 16) After rank pivot1 column1 normalized entry1 = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore pivot row=1;pivot column=1;nonzero pivot=16;old row=[0, 16] After normalized full row=[0, 1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rationals.The inverse operation multiplies back;do not approximate by floats. Step identifiers and dependenciesRule: rank_nonzero_pivot_entire_row_scaling · Uses: G001, S005 S007 · calculationBefore rank eliminate row0 pivot1 column1 entry0 product := mul(15, 0) After rank eliminate row0 pivot1 column1 entry0 product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore rank eliminate row0 pivot1 column1 entry0 difference := sub(1, 0) After rank eliminate row0 pivot1 column1 entry0 difference = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore rank eliminate row0 pivot1 column1 entry1 product := mul(15, 1) After rank eliminate row0 pivot1 column1 entry1 product = 15 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore rank eliminate row0 pivot1 column1 entry1 difference := sub(15, 15) After rank eliminate row0 pivot1 column1 entry1 difference = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore target row=0;old row=[1, 15];subtract factor=15 times pivot row=[0, 1] After new row=[1, 0];cleared pivot-column entry=0 Conditions: none Subtract the factor times the pivot entry in every coordinate;adding back reverses the operation.Clear this pivot column in all other rows before continuing. Step identifiers and dependenciesRule: rank_exact_all_entry_elimination_not_partial_trace · Uses: G001, S010 S012 · impliesBefore complete original square matrix=[[1, 15], [0, 16]];final exact RREF=[[1, 0], [0, 1]];increasing pivot columns=[0, 1] After all rows retained inclzero;rank=2;row operations invertible;row-rank equals column-rank by the finite-dimensional theorem Conditions: none Keep every row and complete all available pivots for exact rank.Invertible row operations preserve rank.Row-rank equals column-rank by the finite-dimensional theorem,not a few successful samples;literal column spans need not remain unchanged. Step identifiers and dependenciesRule: rank_full_rref_rank_preserving_certificate_not_samples · Uses: G001, S011 S013 · calculationBefore whole original matrix diagonal product := mul(1, 16) After whole original matrix diagonal product = 16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore whole original matrix off-diagonal product := mul(15, 0) After whole original matrix off-diagonal product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore whole original matrix determinant difference := sub(16, 0) After whole original matrix determinant difference = 16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore original row indices in SUPPLIED order=[0];original column indices in SUPPLIED order=[0];same whole matrix=[[1, 15], [0, 16]] After selected exact square minor=[[1]] Conditions: none Indices must be distinct but need not be increasing.Extract the minor in supplied order;a permutation can change its determinant sign.Do not use semantic-template sorting to rewrite the question. Step identifiers and dependenciesRule: rank_selected_minor_order_not_semantic_normalization · Uses: G001, S015 S017 · impliesBefore selected supplied-order minor:same1x1 minor=[[1]] After determinant is its sole entry=1 Conditions: none A1x1 determinant is its one original entry,not an empty-minor convention or another cell. Step identifiers and dependenciesRule: rank_one_by_one_minor_entry · Uses: G001, S016 S018 · impliesBefore supplied minor size=1;exact signed determinant=1;nonzero=Yes After certificate from THIS minor=rank_at_least_1 Conditions: none A nonzero r-minor gives rank≥r.A zero selected minor withholds only its certificate;it does not give whole-matrix rank0 or authorize replacing this output with the separate full-RREF result. Step identifiers and dependenciesRule: rank_minor_lower_bound_not_whole_rank_from_zero · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute whole original matrix determinant difference with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute whole original matrix off-diagonal product with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute whole original matrix diagonal product with exact rational arithmetic, not rounded decimals. - 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 rank eliminate row0 pivot1 column1 entry1 difference with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute rank eliminate row0 pivot1 column1 entry1 product with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute rank eliminate row0 pivot1 column1 entry0 difference with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute rank eliminate row0 pivot1 column1 entry0 product with exact rational arithmetic, not rounded decimals. - 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 419d5acd4560fd39f00d34f92f360b9a29682754d24a09f4bd06063f4f156367 - candidate record SHA-256: a4b47623ec0e62e522051e114715b0f4ebb574ae25429493ddb1cf94710af1ce - semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336 - question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261103-r01-matrix-rank-0041.json)