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Question 33125: Addition of Equal-Size Matrices

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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R0065-33125 · q-20261006-advance-20261007-r01-matrix-0075

Original question and givens

Over R, A=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] and B=[[-12, -11, -10, -9, -8], [-14, -13, -12, -11, -10], [-16, -15, -14, -13, -12], [-18, -17, -16, -15, -14], [-20, -19, -18, -17, -16]] have the same dimensions. Find A+B as a nested list of rows.

  • G001 — Over R, A=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] and B=[[-12, -11, -10, -9, -8], [-14, -13, -12, -11, -10], [-16, -15, -14, -13, -12], [-18, -17, -16, -15, -14], [-20, -19, -18, -17, -16]] have the same dimensions. Find A+B 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-addition, matrix_addition, 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 = [A ORIGINAL, [[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]]]

After

derived = [5, 5]

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 = [B ORIGINAL, [[-12, -11, -10, -9, -8], [-14, -13, -12, -11, -10], [-16, -15, -14, -13, -12], [-18, -17, -16, -15, -14], [-20, -19, -18, -17, -16]]]

After

derived = [5, 5]

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, S002

S004 · calculation

Before

Matrix sum ORIGINAL entry0,0 := sum(12, -12)

After

Matrix sum ORIGINAL entry0,0 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Matrix sum ORIGINAL entry0,1 := sum(11, -11)

After

Matrix sum ORIGINAL entry0,1 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Matrix sum ORIGINAL entry0,2 := sum(10, -10)

After

Matrix sum ORIGINAL entry0,2 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Matrix sum ORIGINAL entry0,3 := sum(9, -9)

After

Matrix sum ORIGINAL entry0,3 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Matrix sum ORIGINAL entry0,4 := sum(8, -8)

After

Matrix sum ORIGINAL entry0,4 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Matrix sum ORIGINAL entry1,0 := sum(14, -14)

After

Matrix sum ORIGINAL entry1,0 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Matrix sum ORIGINAL entry1,1 := sum(13, -13)

After

Matrix sum ORIGINAL entry1,1 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Matrix sum ORIGINAL entry1,2 := sum(12, -12)

After

Matrix sum ORIGINAL entry1,2 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Matrix sum ORIGINAL entry1,3 := sum(11, -11)

After

Matrix sum ORIGINAL entry1,3 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Matrix sum ORIGINAL entry1,4 := sum(10, -10)

After

Matrix sum ORIGINAL entry1,4 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Matrix sum ORIGINAL entry2,0 := sum(16, -16)

After

Matrix sum ORIGINAL entry2,0 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Matrix sum ORIGINAL entry2,1 := sum(15, -15)

After

Matrix sum ORIGINAL entry2,1 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Matrix sum ORIGINAL entry2,2 := sum(14, -14)

After

Matrix sum ORIGINAL entry2,2 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Matrix sum ORIGINAL entry2,3 := sum(13, -13)

After

Matrix sum ORIGINAL entry2,3 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Matrix sum ORIGINAL entry2,4 := sum(12, -12)

After

Matrix sum ORIGINAL entry2,4 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Matrix sum ORIGINAL entry3,0 := sum(18, -18)

After

Matrix sum ORIGINAL entry3,0 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Matrix sum ORIGINAL entry3,1 := sum(17, -17)

After

Matrix sum ORIGINAL entry3,1 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Matrix sum ORIGINAL entry3,2 := sum(16, -16)

After

Matrix sum ORIGINAL entry3,2 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Matrix sum ORIGINAL entry3,3 := sum(15, -15)

After

Matrix sum ORIGINAL entry3,3 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Matrix sum ORIGINAL entry3,4 := sum(14, -14)

After

Matrix sum ORIGINAL entry3,4 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Matrix sum ORIGINAL entry4,0 := sum(20, -20)

After

Matrix sum ORIGINAL entry4,0 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Matrix sum ORIGINAL entry4,1 := sum(19, -19)

After

Matrix sum ORIGINAL entry4,1 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Matrix sum ORIGINAL entry4,2 := sum(18, -18)

After

Matrix sum ORIGINAL entry4,2 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Matrix sum ORIGINAL entry4,3 := sum(17, -17)

After

Matrix sum ORIGINAL entry4,3 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Matrix sum ORIGINAL entry4,4 := sum(16, -16)

After

Matrix sum ORIGINAL entry4,4 = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

source roles = [[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]], [[-12, -11, -10, -9, -8], [-14, -13, -12, -11, -10], [-16, -15, -14, -13, -12], [-18, -17, -16, -15, -14], [-20, -19, -18, -17, -16]]]

After

derived = [[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]]

Conditions: none

Add EVERY corresponding entry ONLY at equal shapes.Derived sums may exceed SAFE;do not reparse as a RAW matrix.No zero-padding or broadcasting.

Step identifiers and dependencies

Rule: Matrix_ADDITION_all_corresponding_entries_NO_padding_or_raw_recap · Uses: G001, S028

Reverse checks

  • S029 · 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.
  • S028 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry4,4 with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry4,3 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry4,2 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry4,1 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry4,0 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,4 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,3 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,2 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,1 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,0 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,4 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,3 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,2 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,1 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,0 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,4 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,3 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,2 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,4 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,3 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,0 with exact rational arithmetic, not rounded decimals.
  • 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

[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 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: 31663b645913086eb683cf1300bd820d5e6c05c6a6e4ee75a078578e5c1a2f19
  • candidate record SHA-256: 283acde8a05776265e209534d502d85ada3e91453400bf601b7e1c3dd4251ffe
  • semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
  • question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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