Question 7631: 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=[[-7], [-6]], N=[[0], [-1]]. 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=[[-7], [-6]], N=[[0], [-1]]. 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=[[-7], [-6]];shape=[2, 1]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=-7

target(1,1)=-7

M transpose:source(2,1)=-6

target(1,2)=-6

M shape=[2, 1];N shape=[2, 1]

same shape;entrywise sum permitted

M+N(1,1) := sum(-7, 0)

M+N(1,1) = -7

M+N(2,1) := sum(-6, -1)

M+N(2,1) = -7

sum transpose:source(1,1)=-7

target(1,1)=-7

sum transpose:source(2,1)=-7

target(1,2)=-7

N transpose:source(1,1)=0

target(1,1)=0

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

target(1,2)=-1

MT+NT(1,1) := sum(-7, 0)

MT+NT(1,1) = -7

MT+NT(1,2) := sum(-6, -1)

MT+NT(1,2) = -7

left=[[-7, -7]];right=[[-7, -7]]

same value at every transposed coordinate

Reverse checks

• 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 MT+NT(1,2) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals.
• 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 — Recompute M+N(2,1) with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals.
• 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

([2, 1], [[-7], [-7]], [[-7, -7]], [[-7, -7]])

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.