# Question 7334: Both Matrix Product Orders, Determinants, and Equality [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-0059) 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. R0013-07334 · q-20261012-advance-20261016-r04-determinant-0059 ## 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=[[-4, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?). - 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=[[-4, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?). ## 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 [[-4, 1], [0, 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(-4, 1) After M diagonal product = -4 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, 0) After M off-diagonal product = 0 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(-4, 0) After M signed determinant = -4 Conditions: none Subtract the terms,retaining the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore MN entry0,0 term0 := mul(-4, 1) After MN entry0,0 term0 = -4 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore MN entry0,0 term1 := mul(1, 2) After MN entry0,0 term1 = 2 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore MN entry0,0 := sum(-4, 2) After MN entry0,0 = -2 Conditions: none Add the two row-column products to obtain this cell. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore product=MN;row=0;column=0;terms=[-4, 2] After entry=-2 Conditions: none Retain row/column positions;do not interchange MN and NM or use entrywise multiplication. Step identifiers and dependenciesRule: row_column_matrix_product_cell · Uses: G001, S007 S009 · calculationBefore MN entry0,1 term0 := mul(-4, 0) After MN entry0,1 term0 = 0 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore MN entry0,1 term1 := mul(1, 1) After MN entry0,1 term1 = 1 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore MN entry0,1 := sum(0, 1) After MN entry0,1 = 1 Conditions: none Add the two row-column products to obtain this cell. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore product=MN;row=0;column=1;terms=[0, 1] After entry=1 Conditions: none Retain row/column positions;do not interchange MN and NM or use entrywise multiplication. Step identifiers and dependenciesRule: row_column_matrix_product_cell · Uses: G001, S011 S013 · calculationBefore MN entry1,0 term0 := mul(0, 1) After MN entry1,0 term0 = 0 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore MN entry1,0 term1 := mul(1, 2) After MN entry1,0 term1 = 2 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore MN entry1,0 := sum(0, 2) After MN entry1,0 = 2 Conditions: none Add the two row-column products to obtain this cell. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore product=MN;row=1;column=0;terms=[0, 2] After entry=2 Conditions: none Retain row/column positions;do not interchange MN and NM or use entrywise multiplication. Step identifiers and dependenciesRule: row_column_matrix_product_cell · Uses: G001, S015 S017 · calculationBefore MN entry1,1 term0 := mul(0, 0) After MN entry1,1 term0 = 0 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore MN entry1,1 term1 := mul(1, 1) After MN entry1,1 term1 = 1 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore MN entry1,1 := sum(0, 1) After MN entry1,1 = 1 Conditions: none Add the two row-column products to obtain this cell. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore product=MN;row=1;column=1;terms=[0, 1] After entry=1 Conditions: none Retain row/column positions;do not interchange MN and NM or use entrywise multiplication. Step identifiers and dependenciesRule: row_column_matrix_product_cell · Uses: G001, S019 S021 · calculationBefore NM entry0,0 term0 := mul(1, -4) After NM entry0,0 term0 = -4 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore NM entry0,0 term1 := mul(0, 0) After NM entry0,0 term1 = 0 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore NM entry0,0 := sum(-4, 0) After NM entry0,0 = -4 Conditions: none Add the two row-column products to obtain this cell. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore product=NM;row=0;column=0;terms=[-4, 0] After entry=-4 Conditions: none Retain row/column positions;do not interchange MN and NM or use entrywise multiplication. Step identifiers and dependenciesRule: row_column_matrix_product_cell · Uses: G001, S023 S025 · calculationBefore NM entry0,1 term0 := mul(1, 1) After NM entry0,1 term0 = 1 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore NM entry0,1 term1 := mul(0, 1) After NM entry0,1 term1 = 0 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore NM entry0,1 := sum(1, 0) After NM entry0,1 = 1 Conditions: none Add the two row-column products to obtain this cell. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore product=NM;row=0;column=1;terms=[1, 0] After entry=1 Conditions: none Retain row/column positions;do not interchange MN and NM or use entrywise multiplication. Step identifiers and dependenciesRule: row_column_matrix_product_cell · Uses: G001, S027 S029 · calculationBefore NM entry1,0 term0 := mul(2, -4) After NM entry1,0 term0 = -8 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore NM entry1,0 term1 := mul(1, 0) After NM entry1,0 term1 = 0 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore NM entry1,0 := sum(-8, 0) After NM entry1,0 = -8 Conditions: none Add the two row-column products to obtain this cell. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore product=NM;row=1;column=0;terms=[-8, 0] After entry=-8 Conditions: none Retain row/column positions;do not interchange MN and NM or use entrywise multiplication. Step identifiers and dependenciesRule: row_column_matrix_product_cell · Uses: G001, S031 S033 · calculationBefore NM entry1,1 term0 := mul(2, 1) After NM entry1,1 term0 = 2 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore NM entry1,1 term1 := mul(1, 1) After NM entry1,1 term1 = 1 Conditions: none Multiply the matching k-components of left row i and right column j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore NM entry1,1 := sum(2, 1) After NM entry1,1 = 3 Conditions: none Add the two row-column products to obtain this cell. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · impliesBefore product=NM;row=1;column=1;terms=[2, 1] After entry=3 Conditions: none Retain row/column positions;do not interchange MN and NM or use entrywise multiplication. Step identifiers and dependenciesRule: row_column_matrix_product_cell · Uses: G001, S035 S037 · calculationBefore MN diagonal product := mul(-2, 1) After MN diagonal product = -2 Conditions: none Multiply the main diagonal entries a·d. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore MN off-diagonal product := mul(1, 2) After MN 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, S037 S039 · calculationBefore MN signed determinant := sub(-2, 2) After MN signed determinant = -4 Conditions: none Subtract the terms,retaining the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore NM diagonal product := mul(-4, 3) After NM diagonal product = -12 Conditions: none Multiply the main diagonal entries a·d. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore NM off-diagonal product := mul(1, -8) After NM off-diagonal product = -8 Conditions: none Multiply the off-diagonal entries b·c separately,retaining subtraction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore NM signed determinant := sub(-12, -8) After NM signed determinant = -4 Conditions: none Subtract the terms,retaining the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore N diagonal product := mul(1, 1) After N diagonal product = 1 Conditions: none Multiply the main diagonal entries a·d. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore N off-diagonal product := mul(0, 2) After N off-diagonal product = 0 Conditions: none Multiply the off-diagonal entries b·c separately,retaining subtraction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore N signed determinant := sub(1, 0) After N signed determinant = 1 Conditions: none Subtract the terms,retaining the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore product of original determinants := mul(-4, 1) After product of original determinants = -4 Conditions: none Multiply the two original signed determinants separately. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · impliesBefore MN=[[-2, 1], [2, 1]];NM=[[-4, 1], [-8, 3]];determinants=(-4,-4) After matrices equal=No;D(M)D(N)=-4 Conditions: none Both product determinants here equal D(M)D(N),but matrix equality still requires every cell to agree.Equal determinants do not imply MN=NM. Step identifiers and dependenciesRule: determinant_product_not_matrix_commutation · Uses: G001, S046 ## Reverse checks - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Recompute product of original determinants with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute N signed determinant with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute N off-diagonal product with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute N diagonal product with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute NM signed determinant with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute NM off-diagonal product with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute NM diagonal product with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute MN signed determinant with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute MN off-diagonal product with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute MN diagonal product with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Recompute NM entry1,1 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute NM entry1,1 term1 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute NM entry1,1 term0 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Recompute NM entry1,0 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute NM entry1,0 term1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute NM entry1,0 term0 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Recompute NM entry0,1 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute NM entry0,1 term1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute NM entry0,1 term0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute NM entry0,0 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute NM entry0,0 term1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute NM entry0,0 term0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute MN entry1,1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute MN entry1,1 term1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute MN entry1,1 term0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute MN entry1,0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute MN entry1,0 term1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute MN entry1,0 term0 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 — Recompute MN entry0,1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute MN entry0,1 term1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute MN entry0,1 term0 with exact rational arithmetic, not rounded decimals. - 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 MN entry0,0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute MN entry0,0 term1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute MN entry0,0 term0 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]], [[-4, 1], [-8, 3]], -4, -4, -4, No) 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: 0fe7ebe9f60469c595555d399f816a4d16d1ff3ea18d0adb03962abecc80e8e1 - candidate record SHA-256: 3256e15d4775f055a45146dff698e9573edaea37003282820971bd761220090e - 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-0059.json)