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Question 33123: 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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R0065-33123 · q-20261006-advance-20261007-r01-matrix-0073

Original question and givens

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

  • G001 — Over R, A=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] and B=[[-19, -17, -15], [-18, -16, -14], [-17, -15, -13], [-16, -14, -12], [-15, -13, -11]] 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, [[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]]]

After

derived = [5, 3]

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, [[-19, -17, -15], [-18, -16, -14], [-17, -15, -13], [-16, -14, -12], [-15, -13, -11]]]

After

derived = [5, 3]

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(10, -19)

After

Matrix sum ORIGINAL entry0,0 = -9

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(9, -17)

After

Matrix sum ORIGINAL entry0,1 = -8

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(8, -15)

After

Matrix sum ORIGINAL entry0,2 = -7

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 entry1,0 := sum(12, -18)

After

Matrix sum ORIGINAL entry1,0 = -6

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 entry1,1 := sum(11, -16)

After

Matrix sum ORIGINAL entry1,1 = -5

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 entry1,2 := sum(10, -14)

After

Matrix sum ORIGINAL entry1,2 = -4

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 entry2,0 := sum(14, -17)

After

Matrix sum ORIGINAL entry2,0 = -3

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 entry2,1 := sum(13, -15)

After

Matrix sum ORIGINAL entry2,1 = -2

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 entry2,2 := sum(12, -13)

After

Matrix sum ORIGINAL entry2,2 = -1

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 entry3,0 := sum(16, -16)

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, S012

S014 · calculation

Before

Matrix sum ORIGINAL entry3,1 := sum(15, -14)

After

Matrix sum ORIGINAL entry3,1 = 1

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 entry3,2 := sum(14, -12)

After

Matrix sum ORIGINAL entry3,2 = 2

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 entry4,0 := sum(18, -15)

After

Matrix sum ORIGINAL entry4,0 = 3

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 entry4,1 := sum(17, -13)

After

Matrix sum ORIGINAL entry4,1 = 4

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 entry4,2 := sum(16, -11)

After

Matrix sum ORIGINAL entry4,2 = 5

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 · implies

Before

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

After

derived = [[[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2], [3, 4, 5]]]

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, S018

Reverse checks

  • S019 · 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.
  • S018 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry4,2 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry4,1 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry4,0 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,2 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,0 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,2 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,0 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,2 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,1 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,0 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

[[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2], [3, 4, 5]]

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: f61c0073f4fa049d59a9de2d5cbd376ce8a2ada69429a951fd44a2bf49f4cff0
  • candidate record SHA-256: 63726dccaf48bd8ba8e0239899081561b52ddae3ac3c00ce22ffcc6d1f31f350
  • semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
  • question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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