Question 17604: Zero Determinant Does Not Determine Which Lower Rank Occurs

Original question and givens

Use the complete ordered integer matrix A=[[1, 4], [2, 8]], 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. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.

• G001 — Use the complete ordered integer matrix A=[[1, 4], [2, 8]], 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. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.

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(4, 1)

rank pivot0 column0 normalized entry1 = 4

pivot row=0;pivot column=0;nonzero pivot=1;old row=[1, 4]

normalized full row=[1, 4]

rank eliminate row1 pivot0 column0 entry0 product := mul(2, 1)

rank eliminate row1 pivot0 column0 entry0 product = 2

rank eliminate row1 pivot0 column0 entry0 difference := sub(2, 2)

rank eliminate row1 pivot0 column0 entry0 difference = 0

rank eliminate row1 pivot0 column0 entry1 product := mul(2, 4)

rank eliminate row1 pivot0 column0 entry1 product = 8

rank eliminate row1 pivot0 column0 entry1 difference := sub(8, 8)

rank eliminate row1 pivot0 column0 entry1 difference = 0

target row=1;old row=[2, 8];subtract factor=2 times pivot row=[1, 4]

new row=[0, 0];cleared pivot-column entry=0

whole matrix column=1;unused row indices=[1];remaining column values=[0]

no nonzero pivot;current pivot count=1

complete original square matrix=[[1, 4], [2, 8]];final exact RREF=[[1, 4], [0, 0]];increasing pivot columns=[0]

all rows retained inclzero;rank=1;row operations invertible;row-rank equals column-rank by the finite-dimensional theorem

whole original matrix diagonal product := mul(1, 8)

whole original matrix diagonal product = 8

whole original matrix off-diagonal product := mul(4, 2)

whole original matrix off-diagonal product = 8

whole original matrix determinant difference := sub(8, 8)

whole original matrix determinant difference = 0

exact determinant=0;det-zero premise=Yes;SEPARATE full RREF rank=1

consequence=det_zero_only_excludes_full_rank;zero_determinant_does_not_determine_exact_rank

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute whole original matrix determinant difference with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute whole original matrix off-diagonal product with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute whole original matrix diagonal product with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute rank eliminate row1 pivot0 column0 entry1 difference with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute rank eliminate row1 pivot0 column0 entry1 product with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute rank eliminate row1 pivot0 column0 entry0 difference with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute rank eliminate row1 pivot0 column0 entry0 product 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, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)

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.