# Question 31486: Kernel Membership for a Candidate Scalar [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-0061) 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. R0061-31486 · q-20261006-advance-20261007-r04-eigenvalue-0061 ## Original question and givens Over R, the 2×2 matrix is M=[[-2, 0], [0, 12]]. For the candidate scalar λ=-2 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue. - G001 — Over R, the 2×2 matrix is M=[[-2, 0], [0, 12]]. For the candidate scalar λ=-2 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue. ## 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-kernel-membership, kernel_membership, domain, reals] After derived = [[equality_including_zero_vector_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 dependenciesRule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001 S002 · calculationBefore Eigen ORIGINAL row0 term0 := mul(-2, 1) After Eigen ORIGINAL row0 term0 = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Eigen ORIGINAL row0 term1 := mul(0, 1) After Eigen ORIGINAL row0 term1 = 0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Eigen ORIGINAL Mv row0 sum := sum(-2, 0) After Eigen ORIGINAL Mv row0 sum = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Eigen ORIGINAL lambda times v row0 := mul(-2, 1) After Eigen ORIGINAL lambda times v row0 = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Eigen ORIGINAL residual row0 := sub(-2, -2) After Eigen ORIGINAL residual row0 = 0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Eigen ORIGINAL row1 term0 := mul(0, 1) After Eigen ORIGINAL row1 term0 = 0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Eigen ORIGINAL row1 term1 := mul(12, 1) After Eigen ORIGINAL row1 term1 = 12 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Eigen ORIGINAL Mv row1 sum := sum(0, 12) After Eigen ORIGINAL Mv row1 sum = 12 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Eigen ORIGINAL lambda times v row1 := mul(-2, 1) After Eigen ORIGINAL lambda times v row1 = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Eigen ORIGINAL residual row1 := sub(12, -2) After Eigen ORIGINAL residual row1 = 14 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore source roles = [[[-2, 0], [0, 12]], -2, [1, 1], kernel_membership] After derived = [[-2, 12], [-2, -2], [0, 14], No, Yes, No] Conditions: none Check Mv=λv in EVERY row,then SEPARATELY test nonzero v.An eigenpair needs BOTH;kernel membership needs only(M−λI)v=0,so the zero vector is always in the kernel WITHOUT establishing that λ is an eigenvalue. Step identifiers and dependenciesRule: Eigen_ALL_rows_equation_AND_NONZERO_eigenvector_vs_kernel_guard · Uses: G001, S011 ## Reverse checks - S012 · 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. - S011 · premise_and_result_audit — Recompute Eigen ORIGINAL residual row1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Eigen ORIGINAL lambda times v row1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Eigen ORIGINAL Mv row1 sum with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Eigen ORIGINAL row1 term1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Eigen ORIGINAL row1 term0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Eigen ORIGINAL residual row0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Eigen ORIGINAL lambda times v row0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Eigen ORIGINAL Mv row0 sum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Eigen ORIGINAL row0 term1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Eigen ORIGINAL row0 term0 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 No 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: faf1c2301581e3dd89faeb4630a127f9f59c3b1cf33cbba0a0b49631c8a44c97 - candidate record SHA-256: 3c89d4e57c81f1810794461aa9e53b4bb70bda694639fa1f2e2afe92e7cc1ece - semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69 - question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495 - 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-0061.json)