# Question 7638: 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-0038) 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. R0014-07638 · q-20261012-advance-20261016-r02-matrix-0038 ## 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=[[0, -1, -2], [1, 0, -1], [2, 1, 0]], N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0]]. 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=[[0, -1, -2], [1, 0, -1], [2, 1, 0]], N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0]]. 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=[[0, -1, -2], [1, 0, -1], [2, 1, 0]];shape=[3, 3] 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)=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, 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)=-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, S003 S005 · impliesBefore M 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, S004 S006 · impliesBefore M 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, S005 S007 · 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, S006 S008 · impliesBefore M 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, S007 S009 · impliesBefore M 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, S008 S010 · impliesBefore M 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, S009 S011 · impliesBefore M shape=[3, 3];N shape=[3, 3] 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, S010 S012 · calculationBefore M+N(1,1) := sum(0, 0) After M+N(1,1) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore M+N(1,2) := sum(-1, 1) After M+N(1,2) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore M+N(1,3) := sum(-2, 2) After M+N(1,3) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore M+N(2,1) := sum(1, -1) After M+N(2,1) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore M+N(2,2) := sum(0, 0) After M+N(2,2) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore M+N(2,3) := sum(-1, 1) After M+N(2,3) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore M+N(3,1) := sum(2, -2) After M+N(3,1) = 0 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(3,2) := sum(1, -1) After M+N(3,2) = 0 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(3,3) := sum(0, 0) After M+N(3,3) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore sum 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, S020 S022 · impliesBefore sum transpose:source(1,2)=0 After target(2,1)=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, S021 S023 · impliesBefore sum 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, S022 S024 · impliesBefore sum transpose:source(2,1)=0 After target(1,2)=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, S023 S025 · impliesBefore sum 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, S024 S026 · impliesBefore sum transpose:source(2,3)=0 After target(3,2)=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, S025 S027 · impliesBefore sum transpose:source(3,1)=0 After target(1,3)=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, S026 S028 · impliesBefore sum transpose:source(3,2)=0 After target(2,3)=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, S027 S029 · impliesBefore sum 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, S028 S030 · 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, S029 S031 · 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, S030 S032 · 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, S031 S033 · 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, S032 S034 · 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, S033 S035 · 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, S034 S036 · 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, S035 S037 · 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, S036 S038 · 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, S037 S039 · calculationBefore MT+NT(1,1) := sum(0, 0) After MT+NT(1,1) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore MT+NT(1,2) := sum(1, -1) After MT+NT(1,2) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore MT+NT(1,3) := sum(2, -2) After MT+NT(1,3) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore MT+NT(2,1) := sum(-1, 1) After MT+NT(2,1) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore MT+NT(2,2) := sum(0, 0) After MT+NT(2,2) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore MT+NT(2,3) := sum(1, -1) After MT+NT(2,3) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore MT+NT(3,1) := sum(-2, 2) After MT+NT(3,1) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore MT+NT(3,2) := sum(-1, 1) After MT+NT(3,2) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore MT+NT(3,3) := sum(0, 0) After MT+NT(3,3) = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · impliesBefore left=[[0, 0, 0], [0, 0, 0], [0, 0, 0]];right=[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 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, S047 ## Reverse checks - 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 — Recompute MT+NT(3,3) with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute MT+NT(3,2) with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute MT+NT(3,1) with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute MT+NT(2,3) with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute MT+NT(2,2) with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute MT+NT(2,1) with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute MT+NT(1,3) with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute MT+NT(1,2) with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute M+N(3,3) with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute M+N(3,2) with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute M+N(3,1) with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute M+N(2,3) with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute M+N(2,2) with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute M+N(2,1) with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute M+N(1,3) with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute M+N(1,2) with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals. - 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, 3], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 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: 032b8cd2dd7a0820db7a416bba8050e318b489da0146c18669bb9512c69f4ba7 - candidate record SHA-256: 0e55b7ac20dddd15166f8f39f7688735aaa7f6ceb9e6efb5139ab20072ebc9e7 - 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-0038.json)