Question 17638: The Complete Exact RREF Supplies Matrix Rank

Original question and givens

Use the complete ordered integer matrix A=[[1, 12], [0, 13]], 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 (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.

• G001 — Use the complete ordered integer matrix A=[[1, 12], [0, 13]], 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 (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.

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(12, 1)

rank pivot0 column0 normalized entry1 = 12

pivot row=0;pivot column=0;nonzero pivot=1;old row=[1, 12]

normalized full row=[1, 12]

rank pivot1 column1 normalized entry0 := div(0, 13)

rank pivot1 column1 normalized entry0 = 0

rank pivot1 column1 normalized entry1 := div(13, 13)

rank pivot1 column1 normalized entry1 = 1

pivot row=1;pivot column=1;nonzero pivot=13;old row=[0, 13]

normalized full row=[0, 1]

rank eliminate row0 pivot1 column1 entry0 product := mul(12, 0)

rank eliminate row0 pivot1 column1 entry0 product = 0

rank eliminate row0 pivot1 column1 entry0 difference := sub(1, 0)

rank eliminate row0 pivot1 column1 entry0 difference = 1

rank eliminate row0 pivot1 column1 entry1 product := mul(12, 1)

rank eliminate row0 pivot1 column1 entry1 product = 12

rank eliminate row0 pivot1 column1 entry1 difference := sub(12, 12)

rank eliminate row0 pivot1 column1 entry1 difference = 0

target row=0;old row=[1, 12];subtract factor=12 times pivot row=[0, 1]

new row=[1, 0];cleared pivot-column entry=0

complete original square matrix=[[1, 12], [0, 13]];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, 13)

whole original matrix diagonal product = 13

whole original matrix off-diagonal product := mul(12, 0)

whole original matrix off-diagonal product = 0

whole original matrix determinant difference := sub(13, 0)

whole original matrix determinant difference = 13

Reverse checks

• 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

([[1, 12], [0, 13]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)

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.