# Question 31228: Invertibility over the Real Field [Return to catalogue page 157](https://ocme.evemisslab.com/en/questions/page/157/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261006-advance-20261007-r03-determinant-foundations/#q-20261006-advance-20261007-r03-determinant-0028) 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. R0061-31228 · q-20261006-advance-20261007-r03-determinant-0028 ## Original question and givens Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. Is M invertible over the real field, without requiring an integer-entry inverse? - G001 — Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. Is M invertible over the real field, without requiring an integer-entry inverse? ## 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 source roles = [qf-determinant-invertibility, invertible, domain, reals] After derived = [[real_invertibility_YesNo]] Conditions: none Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status. Step identifiers and dependenciesRule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001 S002 · calculationBefore Det requested matrix ORIGINAL main diagonal product := mul(0, 0) After Det requested matrix ORIGINAL main diagonal product = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Det requested matrix ORIGINAL other diagonal product := mul(1, 1) After Det requested matrix ORIGINAL other diagonal product = 1 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Det requested matrix ORIGINAL signed ad minus bc := sub(0, 1) After Det requested matrix ORIGINAL signed ad minus bc = -1 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [Det requested matrix, [[0, 1], [1, 0]]] After derived = [0, 1, -1] Conditions: none For THIS2×2 matrix explicitly calculate ad and bc,then retain the sign in ad−bc,not the permanent ad+bc.Derived entries are not SAFE-recapped. Step identifiers and dependenciesRule: Determinant_2x2_signed_ad_MINUS_bc · Uses: G001, S004 S006 · impliesBefore source roles = [[[0, 1], [1, 0]], -1] After derived = [Yes] Conditions: none Over the REAL field,a2×2 matrix is invertible IFF its determinant is NONZERO,not only±1.The adjugate divided by det supplies a reverse check;never divide by zero. Step identifiers and dependenciesRule: Determinant_REAL_field_invertible_IFF_NONZERO_not_integer_unit · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S005 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S004 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL signed ad minus bc with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL other diagonal product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL main diagonal product with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. ## Result Yes 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: 3a1ddcaeac429d0b3560f2993d1f2f6dcae09248ce8b278e55080d02727ea8db - candidate record SHA-256: d60c0c7c3f2e2e02c62f7fa214b7c27eab55f7f3614894a3eea5500fd59faccb - semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69 - question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r03-determinant-0028.json)