Question 7680: 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=[[-8, -9, -10, -11, -12]]. 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=[[-8, -9, -10, -11, -12]]. 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=[[-8, -9, -10, -11, -12]];shape=[1, 5]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=-8

target(1,1)=-8

M transpose:source(1,2)=-9

target(2,1)=-9

M transpose:source(1,3)=-10

target(3,1)=-10

M transpose:source(1,4)=-11

target(4,1)=-11

M transpose:source(1,5)=-12

target(5,1)=-12

second transpose:source(1,1)=-8

target(1,1)=-8

second transpose:source(2,1)=-9

target(1,2)=-9

second transpose:source(3,1)=-10

target(1,3)=-10

second transpose:source(4,1)=-11

target(1,4)=-11

second transpose:source(5,1)=-12

target(1,5)=-12

M=[[-8, -9, -10, -11, -12]];M^T=[[-8], [-9], [-10], [-11], [-12]]

roundtrip=[[-8, -9, -10, -11, -12]];equal=Yes

Reverse checks

• 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

([1, 5], [[-8], [-9], [-10], [-11], [-12]], [[-8, -9, -10, -11, -12]], 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.