# Question 3776: Ordered Residual of a Specified Kernel Equation [Return to catalogue page 19](https://ocme.evemisslab.com/en/questions/page/19/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r07-eigenvalue-foundations/#q-20261007-advance-20261009-r07-eigenvalue-0051) 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. R0006-03776 · q-20261007-advance-20261009-r07-eigenvalue-0051 ## Original question and givens Over R, M=[[2, 1], [0, -3]] is 2×2, λ=4, and v=[0, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue. - G001 — Over R, M=[[2, 1], [0, -3]] is 2×2, λ=4, and v=[0, 0]. Give the ordered residual Mv−λv, without assuming 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 [[2, 1], [0, -3]] After exact real2×2 matrix;no spectrum or diagonalization inferred Conditions: none Use the declared real2×2 matrix;coefficients,a discriminant or one candidate residual does not automatically establish the entire spectrum or diagonalizability. Step identifiers and dependenciesRule: two_by_two_matrix_scope · Uses: G001 S002 · calculationBefore row1 product1 := mul(2, 0) After row1 product1 = 0 Conditions: none Multiply corresponding row and vector components without interchanging row and column indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore row1 product2 := mul(1, 0) After row1 product2 = 0 Conditions: none Multiply corresponding row and vector components without interchanging row and column indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Mv coordinate1 := sum(0, 0) After Mv coordinate1 = 0 Conditions: none Add the row products to obtain one component of Mv. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore λv coordinate1 := mul(4, 0) After λv coordinate1 = 0 Conditions: none Multiply λ by the vector component at the same index. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore residual coordinate1 := sub(0, 0) After residual coordinate1 = 0 Conditions: none Subtract λv from Mv coordinatewise;zero residual satisfies this equation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore row2 product1 := mul(0, 0) After row2 product1 = 0 Conditions: none Multiply corresponding row and vector components without interchanging row and column indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore row2 product2 := mul(-3, 0) After row2 product2 = 0 Conditions: none Multiply corresponding row and vector components without interchanging row and column indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Mv coordinate2 := sum(0, 0) After Mv coordinate2 = 0 Conditions: none Add the row products to obtain one component of Mv. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore λv coordinate2 := mul(4, 0) After λv coordinate2 = 0 Conditions: none Multiply λ by the vector component at the same index. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore residual coordinate2 := sub(0, 0) After residual coordinate2 = 0 Conditions: none Subtract λv from Mv coordinatewise;zero residual satisfies this equation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore v=[0, 0];residual=(0, 0) After kernel equation is not an eigenvector certificate for v=0 Conditions: none The zero vector has zero residual for every λ;an eigenvector must additionally be nonzero,and a zero kernel residual cannot remove that premise. Step identifiers and dependenciesRule: zero_vector_kernel_boundary · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute residual coordinate2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute λv coordinate2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Mv coordinate2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute row2 product2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute row2 product1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute residual coordinate1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute λv coordinate1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Mv coordinate1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute row1 product2 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute row1 product1 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (0, 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: 0199bfc1b36b0cd664372f1a00dfa1b5a4c036516a86b04a908f288b1efe5886 - candidate record SHA-256: f2d7e5b1c385efd48798e26b04a89aeb0c0df689ce00b4c45f605a33f140daf7 - semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d - question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r07-eigenvalue-0051.json)