# Question 7365: Signed Determinants Before and After a Row Swap [Return to catalogue page 37](https://ocme.evemisslab.com/en/questions/page/37/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r04-determinant-foundations/#q-20261012-advance-20261016-r04-determinant-0015) 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. R0013-07365 · q-20261012-advance-20261016-r04-determinant-0015 ## Original question and givens Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants). - G001 — Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants). ## 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-determinant](https://ocme.evemisslab.com/en/mko/mko-determinant/) ## Step-by-step derivation S001 · impliesBefore [[2, 1], [2, 1]] After real2×2 integer-entry slice;determinants are signed Conditions: none Compute the signed determinant ad−bc,not its absolute value;matrix multiplication uses row-by-column,not entrywise products.This finite audit is not a general formal proof. Step identifiers and dependenciesRule: signed_two_by_two_determinant_scope · Uses: G001 S002 · calculationBefore M diagonal product := mul(2, 1) After M diagonal product = 2 Conditions: none Multiply the main diagonal entries a·d. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore M off-diagonal product := mul(1, 2) After M off-diagonal product = 2 Conditions: none Multiply the off-diagonal entries b·c separately,retaining subtraction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore M signed determinant := sub(2, 2) After M signed determinant = 0 Conditions: none Subtract the terms,retaining the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore row-swapped M diagonal product := mul(2, 1) After row-swapped M diagonal product = 2 Conditions: none Multiply the main diagonal entries a·d. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore row-swapped M off-diagonal product := mul(1, 2) After row-swapped M off-diagonal product = 2 Conditions: none Multiply the off-diagonal entries b·c separately,retaining subtraction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore row-swapped M signed determinant := sub(2, 2) After row-swapped M signed determinant = 0 Conditions: none Subtract the terms,retaining the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore original plus row-swapped determinants := sum(0, 0) After original plus row-swapped determinants = 0 Conditions: none A row swap reverses the sign,so the two determinants sum to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore M=[[2, 1], [2, 1]];swapped rows=[[2, 1], [2, 1]] After signed determinants=(0,0) Conditions: none This swaps rows rather than transposing;when det=0 its negative is still0,which does not replace the sign rule by absolute values. Step identifiers and dependenciesRule: determinant_row_swap_sign · Uses: G001, S008 ## Reverse checks - 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 — Recompute original plus row-swapped determinants with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute row-swapped M signed determinant with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute row-swapped M off-diagonal product with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute row-swapped M diagonal product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute M signed determinant with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute M off-diagonal product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute M diagonal product with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[2, 1], [2, 1]], 0, [[2, 1], [2, 1]], 0, 0) 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: 2f7eec22093cef53f0c1970cb3f17c255f34a2b98f2f31fa37d7b4cf18dcf4f1 - candidate record SHA-256: bb44da85ae6efd185c2ed47ece058442ec37dfdc8f3992b9d67bcdfc5efc294e - semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59 - question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r04-determinant-0015.json)