Question 7690: 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=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]]. 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=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]]. 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=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]];shape=[3, 5]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=2

target(1,1)=2

M transpose:source(1,2)=1

target(2,1)=1

M transpose:source(1,3)=0

target(3,1)=0

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

target(4,1)=-1

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

target(5,1)=-2

M transpose:source(2,1)=3

target(1,2)=3

M transpose:source(2,2)=2

target(2,2)=2

M transpose:source(2,3)=1

target(3,2)=1

M transpose:source(2,4)=0

target(4,2)=0

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

target(5,2)=-1

M transpose:source(3,1)=4

target(1,3)=4

M transpose:source(3,2)=3

target(2,3)=3

M transpose:source(3,3)=2

target(3,3)=2

M transpose:source(3,4)=1

target(4,3)=1

M transpose:source(3,5)=0

target(5,3)=0

second transpose:source(1,1)=2

target(1,1)=2

second transpose:source(1,2)=3

target(2,1)=3

second transpose:source(1,3)=4

target(3,1)=4

second transpose:source(2,1)=1

target(1,2)=1

second transpose:source(2,2)=2

target(2,2)=2

second transpose:source(2,3)=3

target(3,2)=3

second transpose:source(3,1)=0

target(1,3)=0

second transpose:source(3,2)=1

target(2,3)=1

second transpose:source(3,3)=2

target(3,3)=2

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

target(1,4)=-1

second transpose:source(4,2)=0

target(2,4)=0

second transpose:source(4,3)=1

target(3,4)=1

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

target(1,5)=-2

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

target(2,5)=-1

second transpose:source(5,3)=0

target(3,5)=0

M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]];M^T=[[2, 3, 4], [1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]]

roundtrip=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]];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

([3, 5], [[2, 3, 4], [1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], 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.