# Question 31253: Determinant of a Matrix Product [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-0053) 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. R0061-31253 · q-20261006-advance-20261007-r03-determinant-0053 ## Original question and givens Over R, A=[[0, 1], [1, 0]] and B=[[3, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product. - G001 — Over R, A=[[0, 1], [1, 0]] and B=[[3, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product. ## 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-product, product_determinant, domain, reals] After derived = [[integer_det_AB]] 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 AB row0 column0 term0 := mul(0, 3) After Det AB row0 column0 term0 = 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 AB row0 column0 term1 := mul(1, 2) After Det AB row0 column0 term1 = 2 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 AB row0 column0 sum := sum(0, 2) After Det AB row0 column0 sum = 2 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 · calculationBefore Det AB row0 column1 term0 := mul(0, -1) After Det AB row0 column1 term0 = 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, S004 S006 · calculationBefore Det AB row0 column1 term1 := mul(1, 2) After Det AB row0 column1 term1 = 2 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, S005 S007 · calculationBefore Det AB row0 column1 sum := sum(0, 2) After Det AB row0 column1 sum = 2 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, S006 S008 · calculationBefore Det AB row1 column0 term0 := mul(1, 3) After Det AB row1 column0 term0 = 3 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, S007 S009 · calculationBefore Det AB row1 column0 term1 := mul(0, 2) After Det AB row1 column0 term1 = 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, S008 S010 · calculationBefore Det AB row1 column0 sum := sum(3, 0) After Det AB row1 column0 sum = 3 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, S009 S011 · calculationBefore Det AB row1 column1 term0 := mul(1, -1) After Det AB row1 column1 term0 = -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, S010 S012 · calculationBefore Det AB row1 column1 term1 := mul(0, 2) After Det AB row1 column1 term1 = 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, S011 S013 · calculationBefore Det AB row1 column1 sum := sum(-1, 0) After Det AB row1 column1 sum = -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, S012 S014 · impliesBefore source roles = [[[0, 1], [1, 0]], [[3, -1], [2, 2]]] After derived = [[[2, 2], [3, -1]]] Conditions: none Compute EACH AB entry by A-row times B-column,retaining matrix order;then take THIS AB determinant.This is not entrywise multiplication and does not assert AB=BA. Step identifiers and dependenciesRule: Determinant_PRODUCT_actual_row_by_column_AB_not_Hadamard · Uses: G001, S013 S015 · calculationBefore Det requested matrix ORIGINAL main diagonal product := mul(2, -1) After Det requested matrix ORIGINAL main diagonal product = -2 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, S014 S016 · calculationBefore Det requested matrix ORIGINAL other diagonal product := mul(2, 3) After Det requested matrix ORIGINAL other diagonal product = 6 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, S015 S017 · calculationBefore Det requested matrix ORIGINAL signed ad minus bc := sub(-2, 6) After Det requested matrix ORIGINAL signed ad minus bc = -8 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, S016 S018 · impliesBefore source roles = [Det requested matrix, [[2, 2], [3, -1]]] After derived = [-2, 6, -8] 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, S017 ## Reverse checks - S018 · 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. - S017 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL signed ad minus bc with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL other diagonal product with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL main diagonal product with exact rational arithmetic, not rounded decimals. - S014 · 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. - S013 · premise_and_result_audit — Recompute Det AB row1 column1 sum with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Det AB row1 column1 term1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Det AB row1 column1 term0 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Det AB row1 column0 sum with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Det AB row1 column0 term1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Det AB row1 column0 term0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Det AB row0 column1 sum with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Det AB row0 column1 term1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Det AB row0 column1 term0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Det AB row0 column0 sum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Det AB row0 column0 term1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Det AB row0 column0 term0 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 -8 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: 2e08040d1b07a97ffb05596c13093107eebcb9f8181bf9f5db6429b987840691 - candidate record SHA-256: a9f6d1c45bae43851bfbfefe9a18e854d766420d9e0460c0772861eed6eea607 - 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-0053.json)