OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 7626: Same-Shape Matrix Sum and Both Transpose Routes

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

Return to catalogue page 39

← Question derivation index · Original question batch

R0014-07626 · q-20261012-advance-20261016-r02-matrix-0026

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. Same-shape M=[[0]], N=[[0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

  • 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. Same-shape M=[[0]], N=[[0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

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=[[0]];shape=[1, 1]

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)=0

After

target(1,1)=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, S001

S003 · implies

Before

M shape=[1, 1];N shape=[1, 1]

After

same shape;entrywise sum permitted

Conditions: none

Matrix addition requires equal shapes first.Do not truncate,pad with zeros or broadcast to alter the original problem.

Step identifiers and dependencies

Rule: matrix_sum_shape_guard · Uses: G001, S002

S004 · calculation

Before

M+N(1,1) := sum(0, 0)

After

M+N(1,1) = 0

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

sum transpose:source(1,1)=0

After

target(1,1)=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, S004

S006 · implies

Before

N transpose:source(1,1)=0

After

target(1,1)=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, S005

S007 · calculation

Before

MT+NT(1,1) := sum(0, 0)

After

MT+NT(1,1) = 0

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

left=[[0]];right=[[0]]

After

same value at every transposed coordinate

Conditions: none

One route adds before swapping positions;the other swaps each matrix first and then adds.Both use the same original entry pair.

Step identifiers and dependencies

Rule: transpose_sum_two_routes · Uses: G001, S007

Reverse checks

  • 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 — Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals.
  • 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

([1, 1], [[0]], [[0]], [[0]])

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: 3915f6326a16b73010ce11b2fc232461d6703fad37019c137d7460c3a25928bf
  • candidate record SHA-256: ed62bb6e24f8b3d8158a48c9c6b60ed40c90006e929753d0fd887c316e060128
  • semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
  • question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON