Question 7684: Matrix Shape and Complete Transpose Round Trip

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=[[-4, -5, -6, -7], [-3, -4, -5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

• 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=[[-4, -5, -6, -7], [-3, -4, -5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

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=[[-4, -5, -6, -7], [-3, -4, -5, -6]];shape=[2, 4]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=-4

target(1,1)=-4

M transpose:source(1,2)=-5

target(2,1)=-5

M transpose:source(1,3)=-6

target(3,1)=-6

M transpose:source(1,4)=-7

target(4,1)=-7

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

target(1,2)=-3

M transpose:source(2,2)=-4

target(2,2)=-4

M transpose:source(2,3)=-5

target(3,2)=-5

M transpose:source(2,4)=-6

target(4,2)=-6

second transpose:source(1,1)=-4

target(1,1)=-4

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

target(2,1)=-3

second transpose:source(2,1)=-5

target(1,2)=-5

second transpose:source(2,2)=-4

target(2,2)=-4

second transpose:source(3,1)=-6

target(1,3)=-6

second transpose:source(3,2)=-5

target(2,3)=-5

second transpose:source(4,1)=-7

target(1,4)=-7

second transpose:source(4,2)=-6

target(2,4)=-6

M=[[-4, -5, -6, -7], [-3, -4, -5, -6]];M^T=[[-4, -3], [-5, -4], [-6, -5], [-7, -6]]

roundtrip=[[-4, -5, -6, -7], [-3, -4, -5, -6]];equal=Yes

Reverse checks

• 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 — 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 — 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

([2, 4], [[-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-4, -5, -6, -7], [-3, -4, -5, -6]], Yes)

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.