Question 7646: Same-Shape Matrix Sum and Both Transpose Routes

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=[[8], [9], [10], [11], [12]], N=[[0], [-1], [-2], [-3], [-4]]. 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=[[8], [9], [10], [11], [12]], N=[[0], [-1], [-2], [-3], [-4]]. 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

Step-by-step derivation

M=[[8], [9], [10], [11], [12]];shape=[5, 1]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=8

target(1,1)=8

M transpose:source(2,1)=9

target(1,2)=9

M transpose:source(3,1)=10

target(1,3)=10

M transpose:source(4,1)=11

target(1,4)=11

M transpose:source(5,1)=12

target(1,5)=12

M shape=[5, 1];N shape=[5, 1]

same shape;entrywise sum permitted

M+N(1,1) := sum(8, 0)

M+N(1,1) = 8

M+N(2,1) := sum(9, -1)

M+N(2,1) = 8

M+N(3,1) := sum(10, -2)

M+N(3,1) = 8

M+N(4,1) := sum(11, -3)

M+N(4,1) = 8

M+N(5,1) := sum(12, -4)

M+N(5,1) = 8

sum transpose:source(1,1)=8

target(1,1)=8

sum transpose:source(2,1)=8

target(1,2)=8

sum transpose:source(3,1)=8

target(1,3)=8

sum transpose:source(4,1)=8

target(1,4)=8

sum transpose:source(5,1)=8

target(1,5)=8

N transpose:source(1,1)=0

target(1,1)=0

N transpose:source(2,1)=-1

target(1,2)=-1

N transpose:source(3,1)=-2

target(1,3)=-2

N transpose:source(4,1)=-3

target(1,4)=-3

N transpose:source(5,1)=-4

target(1,5)=-4

MT+NT(1,1) := sum(8, 0)

MT+NT(1,1) = 8

MT+NT(1,2) := sum(9, -1)

MT+NT(1,2) = 8

MT+NT(1,3) := sum(10, -2)

MT+NT(1,3) = 8

MT+NT(1,4) := sum(11, -3)

MT+NT(1,4) = 8

MT+NT(1,5) := sum(12, -4)

MT+NT(1,5) = 8

left=[[8, 8, 8, 8, 8]];right=[[8, 8, 8, 8, 8]]

same value at every transposed coordinate

Reverse checks

• 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 MT+NT(1,5) with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute MT+NT(1,4) with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute MT+NT(1,3) with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute MT+NT(1,2) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute M+N(5,1) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute M+N(4,1) with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute M+N(3,1) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute M+N(2,1) with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals.
• 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

([5, 1], [[8], [8], [8], [8], [8]], [[8, 8, 8, 8, 8]], [[8, 8, 8, 8, 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.