Question 7627: 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=[[-11, -12]], 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=[[-11, -12]], 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=[[-11, -12]];shape=[1, 2]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=-11

target(1,1)=-11

M transpose:source(1,2)=-12

target(2,1)=-12

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

same shape;entrywise sum permitted

M+N(1,1) := sum(-11, 0)

M+N(1,1) = -11

M+N(1,2) := sum(-12, 1)

M+N(1,2) = -11

sum transpose:source(1,1)=-11

target(1,1)=-11

sum transpose:source(1,2)=-11

target(2,1)=-11

N transpose:source(1,1)=0

target(1,1)=0

N transpose:source(1,2)=1

target(2,1)=1

MT+NT(1,1) := sum(-11, 0)

MT+NT(1,1) = -11

MT+NT(2,1) := sum(-12, 1)

MT+NT(2,1) = -11

left=[[-11], [-11]];right=[[-11], [-11]]

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(2,1) 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(1,2) 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

([1, 2], [[-11, -11]], [[-11], [-11]], [[-11], [-11]])

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.