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Question 7689: Matrix Shape and Complete Transpose Round Trip

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0014-07689 · q-20261012-advance-20261016-r02-matrix-0014

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=[[1, 0, -1, -2], [2, 1, 0, -1], [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=[[1, 0, -1, -2], [2, 1, 0, -1], [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.

Step-by-step derivation

S001 · implies

Before

M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]];shape=[3, 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 dependencies

Rule: rectangular_matrix_scope · Uses: G001

S002 · implies

Before

M transpose:source(1,1)=1

After

target(1,1)=1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S001

S003 · implies

Before

M transpose:source(1,2)=0

After

target(2,1)=0

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S002

S004 · implies

Before

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

After

target(3,1)=-1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S003

S005 · implies

Before

M transpose:source(1,4)=-2

After

target(4,1)=-2

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S004

S006 · implies

Before

M transpose:source(2,1)=2

After

target(1,2)=2

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S005

S007 · implies

Before

M transpose:source(2,2)=1

After

target(2,2)=1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S006

S008 · implies

Before

M transpose:source(2,3)=0

After

target(3,2)=0

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S007

S009 · implies

Before

M transpose:source(2,4)=-1

After

target(4,2)=-1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S008

S010 · implies

Before

M transpose:source(3,1)=3

After

target(1,3)=3

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S009

S011 · implies

Before

M transpose:source(3,2)=2

After

target(2,3)=2

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S010

S012 · implies

Before

M transpose:source(3,3)=1

After

target(3,3)=1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S011

S013 · implies

Before

M transpose:source(3,4)=0

After

target(4,3)=0

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S012

S014 · implies

Before

second transpose:source(1,1)=1

After

target(1,1)=1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S013

S015 · implies

Before

second transpose:source(1,2)=2

After

target(2,1)=2

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S014

S016 · implies

Before

second transpose:source(1,3)=3

After

target(3,1)=3

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S015

S017 · implies

Before

second transpose:source(2,1)=0

After

target(1,2)=0

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S016

S018 · implies

Before

second transpose:source(2,2)=1

After

target(2,2)=1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S017

S019 · implies

Before

second transpose:source(2,3)=2

After

target(3,2)=2

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S018

S020 · implies

Before

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

After

target(1,3)=-1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S019

S021 · implies

Before

second transpose:source(3,2)=0

After

target(2,3)=0

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S020

S022 · implies

Before

second transpose:source(3,3)=1

After

target(3,3)=1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S021

S023 · implies

Before

second transpose:source(4,1)=-2

After

target(1,4)=-2

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S022

S024 · implies

Before

second transpose:source(4,2)=-1

After

target(2,4)=-1

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S023

S025 · implies

Before

second transpose:source(4,3)=0

After

target(3,4)=0

Conditions: none

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

Step identifiers and dependencies

Rule: transpose_coordinate_swap · Uses: G001, S024

S026 · implies

Before

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

After

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

Rule: transpose_involution_audit · Uses: G001, S025

Reverse checks

  • 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, 4], [[1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[1, 0, -1, -2], [2, 1, 0, -1], [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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: abd47c1fb893a5c253178343bb1af18a3e72dcf2cf78af79b5cd8ec7c17ca773
  • candidate record SHA-256: f640e1ad6ae503631ccbef5f5d45ea88a4af45abd028ba7bfb091d2288c0ed18
  • semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
  • question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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