# Question 7699: Matrix Shape and Complete Transpose Round Trip

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0014-07699 · q-20261012-advance-20261016-r02-matrix-0024



## 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=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 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=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 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](https://ocme.evemisslab.com/en/mko/mko-matrix/)





## Step-by-step derivation

S001 · impliesBefore

M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]];shape=[5, 4]

After

external indices start at1;transpose is not inverse

Conditions: none

First confirm a nonempty rectangular shape and complete integer entries.Transposition is defined for rectangular matrices without invertibility and is not matrix inversion.

Step identifiers and dependenciesRule: rectangular_matrix_scope · Uses: G001





S002 · impliesBefore

M transpose:source(1,1)=11

After

target(1,1)=11

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S001





S003 · impliesBefore

M transpose:source(1,2)=10

After

target(2,1)=10

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S002





S004 · impliesBefore

M transpose:source(1,3)=9

After

target(3,1)=9

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S003





S005 · impliesBefore

M transpose:source(1,4)=8

After

target(4,1)=8

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S004





S006 · impliesBefore

M transpose:source(2,1)=12

After

target(1,2)=12

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S005





S007 · impliesBefore

M transpose:source(2,2)=11

After

target(2,2)=11

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S006





S008 · impliesBefore

M transpose:source(2,3)=10

After

target(3,2)=10

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S007





S009 · impliesBefore

M transpose:source(2,4)=9

After

target(4,2)=9

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S008





S010 · impliesBefore

M transpose:source(3,1)=13

After

target(1,3)=13

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S009





S011 · impliesBefore

M transpose:source(3,2)=12

After

target(2,3)=12

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S010





S012 · impliesBefore

M transpose:source(3,3)=11

After

target(3,3)=11

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S011





S013 · impliesBefore

M transpose:source(3,4)=10

After

target(4,3)=10

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S012





S014 · impliesBefore

M transpose:source(4,1)=14

After

target(1,4)=14

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S013





S015 · impliesBefore

M transpose:source(4,2)=13

After

target(2,4)=13

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S014





S016 · impliesBefore

M transpose:source(4,3)=12

After

target(3,4)=12

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S015





S017 · impliesBefore

M transpose:source(4,4)=11

After

target(4,4)=11

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S016





S018 · impliesBefore

M transpose:source(5,1)=15

After

target(1,5)=15

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S017





S019 · impliesBefore

M transpose:source(5,2)=14

After

target(2,5)=14

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S018





S020 · impliesBefore

M transpose:source(5,3)=13

After

target(3,5)=13

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S019





S021 · impliesBefore

M transpose:source(5,4)=12

After

target(4,5)=12

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S020





S022 · impliesBefore

second transpose:source(1,1)=11

After

target(1,1)=11

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S021





S023 · impliesBefore

second transpose:source(1,2)=12

After

target(2,1)=12

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S022





S024 · impliesBefore

second transpose:source(1,3)=13

After

target(3,1)=13

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S023





S025 · impliesBefore

second transpose:source(1,4)=14

After

target(4,1)=14

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S024





S026 · impliesBefore

second transpose:source(1,5)=15

After

target(5,1)=15

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S025





S027 · impliesBefore

second transpose:source(2,1)=10

After

target(1,2)=10

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S026





S028 · impliesBefore

second transpose:source(2,2)=11

After

target(2,2)=11

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S027





S029 · impliesBefore

second transpose:source(2,3)=12

After

target(3,2)=12

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S028





S030 · impliesBefore

second transpose:source(2,4)=13

After

target(4,2)=13

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S029





S031 · impliesBefore

second transpose:source(2,5)=14

After

target(5,2)=14

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S030





S032 · impliesBefore

second transpose:source(3,1)=9

After

target(1,3)=9

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S031





S033 · impliesBefore

second transpose:source(3,2)=10

After

target(2,3)=10

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S032





S034 · impliesBefore

second transpose:source(3,3)=11

After

target(3,3)=11

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S033





S035 · impliesBefore

second transpose:source(3,4)=12

After

target(4,3)=12

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S034





S036 · impliesBefore

second transpose:source(3,5)=13

After

target(5,3)=13

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S035





S037 · impliesBefore

second transpose:source(4,1)=8

After

target(1,4)=8

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S036





S038 · impliesBefore

second transpose:source(4,2)=9

After

target(2,4)=9

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S037





S039 · impliesBefore

second transpose:source(4,3)=10

After

target(3,4)=10

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S038





S040 · impliesBefore

second transpose:source(4,4)=11

After

target(4,4)=11

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S039





S041 · impliesBefore

second transpose:source(4,5)=12

After

target(5,4)=12

Conditions: none

Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.

Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S040





S042 · impliesBefore

M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]];M^T=[[11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]]

After

roundtrip=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]];equal=Yes

Conditions: none

The original(i,j) entry moves to(j,i) and then back to(i,j);this is not reversing entries inside a row.

Step identifiers and dependenciesRule: transpose_involution_audit · Uses: G001, S041









## Reverse checks


- S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

([5, 4], [[11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: a5dc9cb5e4e77d939d447335e8413d31ce18aae2665fc7241e6c426537ddddd7
- candidate record SHA-256: 88e6ba8df29272d0116897d99dc5ceda842dccdcadac017088507aa8d470f5df
- semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
- question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r02-matrix-0024.json)