# Question 7641: 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-0041) 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. R0014-07641 · q-20261012-advance-20261016-r02-matrix-0041 ## 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=[[3], [4], [5], [6]], N=[[0], [-1], [-2], [-3]]. 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=[[3], [4], [5], [6]], N=[[0], [-1], [-2], [-3]]. 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=[[3], [4], [5], [6]];shape=[4, 1] 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)=3 After target(1,1)=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, S001 S003 · impliesBefore M transpose:source(2,1)=4 After target(1,2)=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, S002 S004 · impliesBefore M transpose:source(3,1)=5 After target(1,3)=5 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(4,1)=6 After target(1,4)=6 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 shape=[4, 1];N shape=[4, 1] 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, S005 S007 · calculationBefore M+N(1,1) := sum(3, 0) After M+N(1,1) = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore M+N(2,1) := sum(4, -1) After M+N(2,1) = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore M+N(3,1) := sum(5, -2) After M+N(3,1) = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore M+N(4,1) := sum(6, -3) After M+N(4,1) = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore sum transpose:source(1,1)=3 After target(1,1)=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, S010 S012 · impliesBefore sum 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, S011 S013 · impliesBefore sum transpose:source(3,1)=3 After target(1,3)=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 sum transpose:source(4,1)=3 After target(1,4)=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, S013 S015 · 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, S014 S016 · 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, S015 S017 · 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, S016 S018 · impliesBefore N transpose:source(4,1)=-3 After target(1,4)=-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, S017 S019 · calculationBefore MT+NT(1,1) := sum(3, 0) After MT+NT(1,1) = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore MT+NT(1,2) := sum(4, -1) After MT+NT(1,2) = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore MT+NT(1,3) := sum(5, -2) After MT+NT(1,3) = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore MT+NT(1,4) := sum(6, -3) After MT+NT(1,4) = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore left=[[3, 3, 3, 3]];right=[[3, 3, 3, 3]] 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, S022 ## Reverse checks - 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 — Recompute MT+NT(1,4) with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute MT+NT(1,3) with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute MT+NT(1,2) with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute M+N(4,1) with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute M+N(3,1) with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute M+N(2,1) with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals. - 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 ([4, 1], [[3], [3], [3], [3]], [[3, 3, 3, 3]], [[3, 3, 3, 3]]) 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: a722cc829aaca3feaa282dad80b14cf067969315f776b2df94018e5644c7822e - candidate record SHA-256: 09329d671b36f7e90e20d7dfdd5b19aa659738a067954a9b20414d52999394dc - 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-0041.json)