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Question 31508: Number of Distinct Real Eigenvalues of a 2×2 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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R0062-31508 · q-20261006-advance-20261007-r04-eigenvalue-0083

Original question and givens

Over R, the 2×2 matrix is M=[[3, -1], [4, 3]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

  • G001 — Over R, the 2×2 matrix is M=[[3, -1], [4, 3]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

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-eigenvalue-real-count, reals, real_eigenvalue_count, no-form-field]

After

derived = [[distinct_real_count_STRING]]

Conditions: none

Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction.

Step identifiers and dependencies

Rule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001

S002 · calculation

Before

Eigen count ORIGINAL diagonal trace := sum(3, 3)

After

Eigen count ORIGINAL diagonal trace = 6

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Eigen count ORIGINAL ad := mul(3, 3)

After

Eigen count ORIGINAL ad = 9

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Eigen count ORIGINAL bc := mul(-1, 4)

After

Eigen count ORIGINAL bc = -4

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Eigen count ORIGINAL ad minus bc := sub(9, -4)

After

Eigen count ORIGINAL ad minus bc = 13

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Eigen count ORIGINAL negative trace := sub(0, 6)

After

Eigen count ORIGINAL negative trace = -6

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Eigen count ORIGINAL trace squared := mul(6, 6)

After

Eigen count ORIGINAL trace squared = 36

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Eigen count ORIGINAL four times determinant := mul(4, 13)

After

Eigen count ORIGINAL four times determinant = 52

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Eigen count ORIGINAL characteristic discriminant := sub(36, 52)

After

Eigen count ORIGINAL characteristic discriminant = -16

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [[[3, -1], [4, 3]]]

After

derived = [[1, -6, 13], -16]

Conditions: none

Expand det(λI−M)=(λ−a)(λ−d)−bc to obtain λ²−(a+d)λ+(ad−bc),then its discriminant.Coefficients/discriminant are exact derived values without a RAW SAFE recap.

Step identifiers and dependencies

Rule: Eigen_COUNT_characteristic_polynomial_ALL_coefficients · Uses: G001, S009

S011 · implies

Before

source roles = [-16]

After

derived = [0]

Conditions: none

A real quadratic has0/1/2 DISTINCT real roots for negative/zero/positive discriminant.Positive nonsquare values still give2;count a repeated root ONCE.No diagonalizability,Jordan or complex-spectrum claim follows.

Step identifiers and dependencies

Rule: Eigen_COUNT_DISTINCT_REAL_roots_by_sign_NOT_square_or_multiplicity · Uses: G001, S010

Reverse checks

  • S011 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S010 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Recompute Eigen count ORIGINAL characteristic discriminant with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Eigen count ORIGINAL four times determinant with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Eigen count ORIGINAL trace squared with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Eigen count ORIGINAL negative trace with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Eigen count ORIGINAL ad minus bc with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Eigen count ORIGINAL bc with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Eigen count ORIGINAL ad with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Eigen count ORIGINAL diagonal trace with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

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: 5e24875cf745d2bbe5ce3b3c1449fecfe9b422f1ed243f5e2d9168a2ec0ce997
  • candidate record SHA-256: 66103ef67ed6fd63868b779bc4c549683dce8e748cfde3a2224066a1d9df7e7c
  • semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd
  • question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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