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Question 31234: Invertibility over the Real Field

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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R0061-31234 · q-20261006-advance-20261007-r03-determinant-0034

Original question and givens

Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. Is M invertible over the real field, without requiring an integer-entry inverse?

  • G001 — Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. Is M invertible over the real field, without requiring an integer-entry inverse?

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-determinant-invertibility, invertible, domain, reals]

After

derived = [[real_invertibility_YesNo]]

Conditions: none

Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status.

Step identifiers and dependencies

Rule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001

S002 · calculation

Before

Det requested matrix ORIGINAL main diagonal product := mul(-4, 9)

After

Det requested matrix ORIGINAL main diagonal product = -36

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Det requested matrix ORIGINAL other diagonal product := mul(1, -2)

After

Det requested matrix ORIGINAL other diagonal product = -2

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Det requested matrix ORIGINAL signed ad minus bc := sub(-36, -2)

After

Det requested matrix ORIGINAL signed ad minus bc = -34

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [Det requested matrix, [[-4, 1], [-2, 9]]]

After

derived = [-36, -2, -34]

Conditions: none

For THIS2×2 matrix explicitly calculate ad and bc,then retain the sign in ad−bc,not the permanent ad+bc.Derived entries are not SAFE-recapped.

Step identifiers and dependencies

Rule: Determinant_2x2_signed_ad_MINUS_bc · Uses: G001, S004

S006 · implies

Before

source roles = [[[-4, 1], [-2, 9]], -34]

After

derived = [Yes]

Conditions: none

Over the REAL field,a2×2 matrix is invertible IFF its determinant is NONZERO,not only±1.The adjugate divided by det supplies a reverse check;never divide by zero.

Step identifiers and dependencies

Rule: Determinant_REAL_field_invertible_IFF_NONZERO_not_integer_unit · Uses: G001, S005

Reverse checks

  • S006 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
  • S005 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
  • S004 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL signed ad minus bc with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL other diagonal product with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL main diagonal product with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.

Result

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: 2811ca5ae911440846ccd8a627f52fa6b49421376051770c25c214cac2d17517
  • candidate record SHA-256: b20e5762b8706b38033f4bf275f6e988476d9bb7deb8dd352ca4f5b7004ef15c
  • semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69
  • question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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