Question 7694: 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=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 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=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 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=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]];shape=[4, 4]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=6

target(1,1)=6

M transpose:source(1,2)=5

target(2,1)=5

M transpose:source(1,3)=4

target(3,1)=4

M transpose:source(1,4)=3

target(4,1)=3

M transpose:source(2,1)=7

target(1,2)=7

M transpose:source(2,2)=6

target(2,2)=6

M transpose:source(2,3)=5

target(3,2)=5

M transpose:source(2,4)=4

target(4,2)=4

M transpose:source(3,1)=8

target(1,3)=8

M transpose:source(3,2)=7

target(2,3)=7

M transpose:source(3,3)=6

target(3,3)=6

M transpose:source(3,4)=5

target(4,3)=5

M transpose:source(4,1)=9

target(1,4)=9

M transpose:source(4,2)=8

target(2,4)=8

M transpose:source(4,3)=7

target(3,4)=7

M transpose:source(4,4)=6

target(4,4)=6

second transpose:source(1,1)=6

target(1,1)=6

second transpose:source(1,2)=7

target(2,1)=7

second transpose:source(1,3)=8

target(3,1)=8

second transpose:source(1,4)=9

target(4,1)=9

second transpose:source(2,1)=5

target(1,2)=5

second transpose:source(2,2)=6

target(2,2)=6

second transpose:source(2,3)=7

target(3,2)=7

second transpose:source(2,4)=8

target(4,2)=8

second transpose:source(3,1)=4

target(1,3)=4

second transpose:source(3,2)=5

target(2,3)=5

second transpose:source(3,3)=6

target(3,3)=6

second transpose:source(3,4)=7

target(4,3)=7

second transpose:source(4,1)=3

target(1,4)=3

second transpose:source(4,2)=4

target(2,4)=4

second transpose:source(4,3)=5

target(3,4)=5

second transpose:source(4,4)=6

target(4,4)=6

M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]];M^T=[[6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]]

roundtrip=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]];equal=Yes

Reverse checks

• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([4, 4], [[6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 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.