Question 7698: 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=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 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=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 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=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]];shape=[5, 3]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=10

target(1,1)=10

M transpose:source(1,2)=9

target(2,1)=9

M transpose:source(1,3)=8

target(3,1)=8

M transpose:source(2,1)=11

target(1,2)=11

M transpose:source(2,2)=10

target(2,2)=10

M transpose:source(2,3)=9

target(3,2)=9

M transpose:source(3,1)=12

target(1,3)=12

M transpose:source(3,2)=11

target(2,3)=11

M transpose:source(3,3)=10

target(3,3)=10

M transpose:source(4,1)=13

target(1,4)=13

M transpose:source(4,2)=12

target(2,4)=12

M transpose:source(4,3)=11

target(3,4)=11

M transpose:source(5,1)=14

target(1,5)=14

M transpose:source(5,2)=13

target(2,5)=13

M transpose:source(5,3)=12

target(3,5)=12

second transpose:source(1,1)=10

target(1,1)=10

second transpose:source(1,2)=11

target(2,1)=11

second transpose:source(1,3)=12

target(3,1)=12

second transpose:source(1,4)=13

target(4,1)=13

second transpose:source(1,5)=14

target(5,1)=14

second transpose:source(2,1)=9

target(1,2)=9

second transpose:source(2,2)=10

target(2,2)=10

second transpose:source(2,3)=11

target(3,2)=11

second transpose:source(2,4)=12

target(4,2)=12

second transpose:source(2,5)=13

target(5,2)=13

second transpose:source(3,1)=8

target(1,3)=8

second transpose:source(3,2)=9

target(2,3)=9

second transpose:source(3,3)=10

target(3,3)=10

second transpose:source(3,4)=11

target(4,3)=11

second transpose:source(3,5)=12

target(5,3)=12

M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]];M^T=[[10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]]

roundtrip=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]];equal=Yes

Reverse checks

• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([5, 3], [[10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 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.