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Question 7700: 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

R0014-07700 · q-20261012-advance-20261016-r02-matrix-0025

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=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 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=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 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.

Step-by-step derivation

S001 · implies

Before

M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]];shape=[5, 5]

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)=12

After

target(1,1)=12

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)=11

After

target(2,1)=11

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)=10

After

target(3,1)=10

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)=9

After

target(4,1)=9

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(1,5)=8

After

target(5,1)=8

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(2,1)=13

After

target(1,2)=13

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(2,2)=12

After

target(2,2)=12

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(2,3)=11

After

target(3,2)=11

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(2,4)=10

After

target(4,2)=10

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(2,5)=9

After

target(5,2)=9

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(3,1)=14

After

target(1,3)=14

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(3,2)=13

After

target(2,3)=13

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

M transpose:source(3,3)=12

After

target(3,3)=12

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

M transpose:source(3,4)=11

After

target(4,3)=11

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

M transpose:source(3,5)=10

After

target(5,3)=10

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

M transpose:source(4,1)=15

After

target(1,4)=15

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

M transpose:source(4,2)=14

After

target(2,4)=14

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

M transpose:source(4,3)=13

After

target(3,4)=13

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

M transpose:source(4,4)=12

After

target(4,4)=12

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

M transpose:source(4,5)=11

After

target(5,4)=11

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

M transpose:source(5,1)=16

After

target(1,5)=16

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

M transpose:source(5,2)=15

After

target(2,5)=15

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

M transpose:source(5,3)=14

After

target(3,5)=14

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

M transpose:source(5,4)=13

After

target(4,5)=13

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 transpose:source(5,5)=12

After

target(5,5)=12

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, S025

S027 · implies

Before

second transpose:source(1,1)=12

After

target(1,1)=12

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, S026

S028 · implies

Before

second transpose:source(1,2)=13

After

target(2,1)=13

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, S027

S029 · implies

Before

second transpose:source(1,3)=14

After

target(3,1)=14

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, S028

S030 · implies

Before

second transpose:source(1,4)=15

After

target(4,1)=15

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, S029

S031 · implies

Before

second transpose:source(1,5)=16

After

target(5,1)=16

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, S030

S032 · implies

Before

second transpose:source(2,1)=11

After

target(1,2)=11

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, S031

S033 · implies

Before

second transpose:source(2,2)=12

After

target(2,2)=12

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, S032

S034 · implies

Before

second transpose:source(2,3)=13

After

target(3,2)=13

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, S033

S035 · implies

Before

second transpose:source(2,4)=14

After

target(4,2)=14

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, S034

S036 · implies

Before

second transpose:source(2,5)=15

After

target(5,2)=15

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, S035

S037 · implies

Before

second transpose:source(3,1)=10

After

target(1,3)=10

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, S036

S038 · implies

Before

second transpose:source(3,2)=11

After

target(2,3)=11

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, S037

S039 · implies

Before

second transpose:source(3,3)=12

After

target(3,3)=12

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, S038

S040 · implies

Before

second transpose:source(3,4)=13

After

target(4,3)=13

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, S039

S041 · implies

Before

second transpose:source(3,5)=14

After

target(5,3)=14

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, S040

S042 · implies

Before

second transpose:source(4,1)=9

After

target(1,4)=9

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, S041

S043 · implies

Before

second transpose:source(4,2)=10

After

target(2,4)=10

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, S042

S044 · implies

Before

second transpose:source(4,3)=11

After

target(3,4)=11

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, S043

S045 · implies

Before

second transpose:source(4,4)=12

After

target(4,4)=12

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, S044

S046 · implies

Before

second transpose:source(4,5)=13

After

target(5,4)=13

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, S045

S047 · implies

Before

second transpose:source(5,1)=8

After

target(1,5)=8

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, S046

S048 · implies

Before

second transpose:source(5,2)=9

After

target(2,5)=9

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, S047

S049 · implies

Before

second transpose:source(5,3)=10

After

target(3,5)=10

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, S048

S050 · implies

Before

second transpose:source(5,4)=11

After

target(4,5)=11

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, S049

S051 · implies

Before

second transpose:source(5,5)=12

After

target(5,5)=12

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, S050

S052 · implies

Before

M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]];M^T=[[12, 13, 14, 15, 16], [11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]]

After

roundtrip=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 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 dependencies

Rule: transpose_involution_audit · Uses: G001, S051

Reverse checks

  • S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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, 5], [[12, 13, 14, 15, 16], [11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 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: c9626b340194021ca25d4a9996df9d3b846534ca2b6679039ecc52aaac489abe
  • candidate record SHA-256: dc2c885c668ca0c3eefcafd45f71f96b4f0b8762828ece9ba20d1aa3db14be19
  • semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
  • question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON