# Question 31514: Number of Distinct Real Eigenvalues of a 2×2 Matrix [Return to catalogue page 158](https://ocme.evemisslab.com/en/questions/page/158/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261006-advance-20261007-r04-eigenvalue-foundations/#q-20261006-advance-20261007-r04-eigenvalue-0089) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0062-31514 · q-20261006-advance-20261007-r04-eigenvalue-0089 ## Original question and givens Over R, the 2×2 matrix is M=[[5, -1], [6, 5]]. 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=[[5, -1], [6, 5]]. 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. - [mko-eigenvalue](https://ocme.evemisslab.com/en/mko/mko-eigenvalue/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001 S002 · calculationBefore Eigen count ORIGINAL diagonal trace := sum(5, 5) After Eigen count ORIGINAL diagonal trace = 10 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Eigen count ORIGINAL ad := mul(5, 5) After Eigen count ORIGINAL ad = 25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Eigen count ORIGINAL bc := mul(-1, 6) After Eigen count ORIGINAL bc = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Eigen count ORIGINAL ad minus bc := sub(25, -6) After Eigen count ORIGINAL ad minus bc = 31 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Eigen count ORIGINAL negative trace := sub(0, 10) After Eigen count ORIGINAL negative trace = -10 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Eigen count ORIGINAL trace squared := mul(10, 10) After Eigen count ORIGINAL trace squared = 100 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Eigen count ORIGINAL four times determinant := mul(4, 31) After Eigen count ORIGINAL four times determinant = 124 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Eigen count ORIGINAL characteristic discriminant := sub(100, 124) After Eigen count ORIGINAL characteristic discriminant = -24 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [[[5, -1], [6, 5]]] After derived = [[1, -10, 31], -24] 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 dependenciesRule: Eigen_COUNT_characteristic_polynomial_ALL_coefficients · Uses: G001, S009 S011 · impliesBefore source roles = [-24] 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 dependenciesRule: 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: 74848e41e1110a5917a1483fd4605cb4539e1e1f4808e1a38595dc463f0d8f2a - candidate record SHA-256: e4c15ad410d41f2675ada238148404a639b4beee17661ca4534131f41bca68c4 - semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd - question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r04-eigenvalue-0089.json)