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Question 33194: Transpose of a Rectangular Matrix

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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R0065-33194 · q-20261006-advance-20261007-r01-matrix-0094

Original question and givens

The real matrix M=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

  • G001 — The real matrix M=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

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

source roles = [qf-matrix-transpose, matrix_transpose, reals, []]

After

derived = [[nested_ROW_matrix]]

Conditions: none

MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status.

Step identifiers and dependencies

Rule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001

S002 · implies

Before

source roles = [M ORIGINAL, [[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]]]

After

derived = [4, 4]

Conditions: none

THIS bank allows1..5 by1..5 matrices,not the R64 2×2 restriction.Type EVERY raw SAFE cell,including unused shape/entry cells.Keep original row order.

Step identifiers and dependencies

Rule: Matrix_RAW_rectangular_1to5_ALL_SAFE_entries_even_unused · Uses: G001, S001

S003 · implies

Before

source roles = [1, 1, 6]

After

derived = [1, 1, 6]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S002

S004 · implies

Before

source roles = [2, 1, 8]

After

derived = [1, 2, 8]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S003

S005 · implies

Before

source roles = [3, 1, 10]

After

derived = [1, 3, 10]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S004

S006 · implies

Before

source roles = [4, 1, 12]

After

derived = [1, 4, 12]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S005

S007 · implies

Before

source roles = [1, 2, 5]

After

derived = [2, 1, 5]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S006

S008 · implies

Before

source roles = [2, 2, 7]

After

derived = [2, 2, 7]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S007

S009 · implies

Before

source roles = [3, 2, 9]

After

derived = [2, 3, 9]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S008

S010 · implies

Before

source roles = [4, 2, 11]

After

derived = [2, 4, 11]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S009

S011 · implies

Before

source roles = [1, 3, 4]

After

derived = [3, 1, 4]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S010

S012 · implies

Before

source roles = [2, 3, 6]

After

derived = [3, 2, 6]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S011

S013 · implies

Before

source roles = [3, 3, 8]

After

derived = [3, 3, 8]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S012

S014 · implies

Before

source roles = [4, 3, 10]

After

derived = [3, 4, 10]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S013

S015 · implies

Before

source roles = [1, 4, 3]

After

derived = [4, 1, 3]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S014

S016 · implies

Before

source roles = [2, 4, 5]

After

derived = [4, 2, 5]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S015

S017 · implies

Before

source roles = [3, 4, 7]

After

derived = [4, 3, 7]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S016

S018 · implies

Before

source roles = [4, 4, 9]

After

derived = [4, 4, 9]

Conditions: none

Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange.

Step identifiers and dependencies

Rule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S017

Reverse checks

  • S018 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S017 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S016 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S015 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S014 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S012 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S011 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S010 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S008 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S007 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S006 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S004 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S003 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.

Result

[[6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]

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: f7a60d722460a896a457db9ff97eb76f9ec329fa096357b62e8df501290ab9d6
  • candidate record SHA-256: fe2ae1797ebd7319322087ee8ba015c1f4dd8923c2d7ae85ffcda4174b2ff5a3
  • semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
  • question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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