Question 7616: Candidate Transpose Shape before Entrywise Residuals

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. M=[[3, 1], [2, -1]], candidate=[[4, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

• 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. M=[[3, 1], [2, -1]], candidate=[[4, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

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=[[3, 1], [2, -1]];shape=[2, 2]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=3

target(1,1)=3

M transpose:source(1,2)=1

target(2,1)=1

M transpose:source(2,1)=2

target(1,2)=2

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

target(2,2)=-1

candidate shape=[2, 2];required transpose shape=[2, 2]

qualified=Yes

candidate-MT(1,1) := sub(4, 3)

candidate-MT(1,1) = 1

candidate-MT(1,2) := sub(2, 2)

candidate-MT(1,2) = 0

candidate-MT(2,1) := sub(1, 1)

candidate-MT(2,1) = 0

candidate-MT(2,2) := sub(-1, -1)

candidate-MT(2,2) = 0

qualified=Yes;residual/status=[[1, 0], [0, 0]]

equal=No

Reverse checks

• 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 candidate-MT(2,2) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute candidate-MT(2,1) with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute candidate-MT(1,2) with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute candidate-MT(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

([[3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

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.