# Question 7640: Same-Shape Matrix Sum and Both Transpose Routes [Return to catalogue page 39](https://ocme.evemisslab.com/en/questions/page/39/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r02-matrix-foundations/#q-20261012-advance-20261016-r02-matrix-0040) 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. R0014-07640 · q-20261012-advance-20261016-r02-matrix-0040 ## Original question and givens Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ). - G001 — Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ). ## 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](https://ocme.evemisslab.com/en/mko/mko-matrix/) ## Step-by-step derivation S001 · impliesBefore M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]];shape=[3, 5] After external indices start at1;transpose is not inverse Conditions: none First confirm a nonempty rectangular shape and complete integer entries.Transposition is defined for rectangular matrices without invertibility and is not matrix inversion. Step identifiers and dependenciesRule: rectangular_matrix_scope · Uses: G001 S002 · impliesBefore M transpose:source(1,1)=2 After target(1,1)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S001 S003 · impliesBefore M transpose:source(1,2)=1 After target(2,1)=1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S002 S004 · impliesBefore M transpose:source(1,3)=0 After target(3,1)=0 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S003 S005 · impliesBefore M transpose:source(1,4)=-1 After target(4,1)=-1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S004 S006 · impliesBefore M transpose:source(1,5)=-2 After target(5,1)=-2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S005 S007 · impliesBefore M transpose:source(2,1)=3 After target(1,2)=3 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S006 S008 · impliesBefore M transpose:source(2,2)=2 After target(2,2)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S007 S009 · impliesBefore M transpose:source(2,3)=1 After target(3,2)=1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S008 S010 · impliesBefore M transpose:source(2,4)=0 After target(4,2)=0 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S009 S011 · impliesBefore M transpose:source(2,5)=-1 After target(5,2)=-1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S010 S012 · impliesBefore M transpose:source(3,1)=4 After target(1,3)=4 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S011 S013 · impliesBefore M transpose:source(3,2)=3 After target(2,3)=3 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S012 S014 · impliesBefore M transpose:source(3,3)=2 After target(3,3)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S013 S015 · impliesBefore M transpose:source(3,4)=1 After target(4,3)=1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S014 S016 · impliesBefore M transpose:source(3,5)=0 After target(5,3)=0 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S015 S017 · impliesBefore M shape=[3, 5];N shape=[3, 5] After same shape;entrywise sum permitted Conditions: none Matrix addition requires equal shapes first.Do not truncate,pad with zeros or broadcast to alter the original problem. Step identifiers and dependenciesRule: matrix_sum_shape_guard · Uses: G001, S016 S018 · calculationBefore M+N(1,1) := sum(2, 0) After M+N(1,1) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore M+N(1,2) := sum(1, 1) After M+N(1,2) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore M+N(1,3) := sum(0, 2) After M+N(1,3) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore M+N(1,4) := sum(-1, 3) After M+N(1,4) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore M+N(1,5) := sum(-2, 4) After M+N(1,5) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore M+N(2,1) := sum(3, -1) After M+N(2,1) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore M+N(2,2) := sum(2, 0) After M+N(2,2) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore M+N(2,3) := sum(1, 1) After M+N(2,3) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore M+N(2,4) := sum(0, 2) After M+N(2,4) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore M+N(2,5) := sum(-1, 3) After M+N(2,5) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore M+N(3,1) := sum(4, -2) After M+N(3,1) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore M+N(3,2) := sum(3, -1) After M+N(3,2) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore M+N(3,3) := sum(2, 0) After M+N(3,3) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore M+N(3,4) := sum(1, 1) After M+N(3,4) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore M+N(3,5) := sum(0, 2) After M+N(3,5) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore sum transpose:source(1,1)=2 After target(1,1)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S032 S034 · impliesBefore sum transpose:source(1,2)=2 After target(2,1)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S033 S035 · impliesBefore sum transpose:source(1,3)=2 After target(3,1)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S034 S036 · impliesBefore sum transpose:source(1,4)=2 After target(4,1)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S035 S037 · impliesBefore sum transpose:source(1,5)=2 After target(5,1)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S036 S038 · impliesBefore sum transpose:source(2,1)=2 After target(1,2)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S037 S039 · impliesBefore sum transpose:source(2,2)=2 After target(2,2)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S038 S040 · impliesBefore sum transpose:source(2,3)=2 After target(3,2)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S039 S041 · impliesBefore sum transpose:source(2,4)=2 After target(4,2)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S040 S042 · impliesBefore sum transpose:source(2,5)=2 After target(5,2)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S041 S043 · impliesBefore sum transpose:source(3,1)=2 After target(1,3)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S042 S044 · impliesBefore sum transpose:source(3,2)=2 After target(2,3)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S043 S045 · impliesBefore sum transpose:source(3,3)=2 After target(3,3)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S044 S046 · impliesBefore sum transpose:source(3,4)=2 After target(4,3)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S045 S047 · impliesBefore sum transpose:source(3,5)=2 After target(5,3)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S046 S048 · impliesBefore N transpose:source(1,1)=0 After target(1,1)=0 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S047 S049 · impliesBefore N transpose:source(1,2)=1 After target(2,1)=1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S048 S050 · impliesBefore N transpose:source(1,3)=2 After target(3,1)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S049 S051 · impliesBefore N transpose:source(1,4)=3 After target(4,1)=3 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S050 S052 · impliesBefore N transpose:source(1,5)=4 After target(5,1)=4 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S051 S053 · impliesBefore N transpose:source(2,1)=-1 After target(1,2)=-1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S052 S054 · impliesBefore N transpose:source(2,2)=0 After target(2,2)=0 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S053 S055 · impliesBefore N transpose:source(2,3)=1 After target(3,2)=1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S054 S056 · impliesBefore N transpose:source(2,4)=2 After target(4,2)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S055 S057 · impliesBefore N transpose:source(2,5)=3 After target(5,2)=3 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S056 S058 · impliesBefore N transpose:source(3,1)=-2 After target(1,3)=-2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S057 S059 · impliesBefore N transpose:source(3,2)=-1 After target(2,3)=-1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S058 S060 · impliesBefore N transpose:source(3,3)=0 After target(3,3)=0 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S059 S061 · impliesBefore N transpose:source(3,4)=1 After target(4,3)=1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S060 S062 · impliesBefore N transpose:source(3,5)=2 After target(5,3)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S061 S063 · calculationBefore MT+NT(1,1) := sum(2, 0) After MT+NT(1,1) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · calculationBefore MT+NT(1,2) := sum(3, -1) After MT+NT(1,2) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore MT+NT(1,3) := sum(4, -2) After MT+NT(1,3) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore MT+NT(2,1) := sum(1, 1) After MT+NT(2,1) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore MT+NT(2,2) := sum(2, 0) After MT+NT(2,2) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore MT+NT(2,3) := sum(3, -1) After MT+NT(2,3) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore MT+NT(3,1) := sum(0, 2) After MT+NT(3,1) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore MT+NT(3,2) := sum(1, 1) After MT+NT(3,2) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore MT+NT(3,3) := sum(2, 0) After MT+NT(3,3) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore MT+NT(4,1) := sum(-1, 3) After MT+NT(4,1) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore MT+NT(4,2) := sum(0, 2) After MT+NT(4,2) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore MT+NT(4,3) := sum(1, 1) After MT+NT(4,3) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · calculationBefore MT+NT(5,1) := sum(-2, 4) After MT+NT(5,1) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore MT+NT(5,2) := sum(-1, 3) After MT+NT(5,2) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore MT+NT(5,3) := sum(0, 2) After MT+NT(5,3) = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · impliesBefore left=[[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]];right=[[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]] After same value at every transposed coordinate Conditions: none One route adds before swapping positions;the other swaps each matrix first and then adds.Both use the same original entry pair. Step identifiers and dependenciesRule: transpose_sum_two_routes · Uses: G001, S077 ## Reverse checks - S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S077 · premise_and_result_audit — Recompute MT+NT(5,3) with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute MT+NT(5,2) with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute MT+NT(5,1) with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute MT+NT(4,3) with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute MT+NT(4,2) with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute MT+NT(4,1) with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute MT+NT(3,3) with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute MT+NT(3,2) with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute MT+NT(3,1) with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute MT+NT(2,3) with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute MT+NT(2,2) with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute MT+NT(2,1) with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute MT+NT(1,3) with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute MT+NT(1,2) with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute M+N(3,5) with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute M+N(3,4) with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute M+N(3,3) with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute M+N(3,2) with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute M+N(3,1) with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute M+N(2,5) with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute M+N(2,4) with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute M+N(2,3) with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute M+N(2,2) with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute M+N(2,1) with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute M+N(1,5) with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute M+N(1,4) with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute M+N(1,3) with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute M+N(1,2) with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([3, 5], [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]]) 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: 5843a8cb1ed130fa0fe7f0c9f95bf9872dda1385118af50bd0208fe1c39dfa86 - candidate record SHA-256: fee8880aca1902474687fa44e0e134cbf09fa584c7cd3e6432b3e23473fbdf78 - semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77 - question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r02-matrix-0040.json)