Question 3772: First Mismatch of a Nonzero Eigenvector Candidate

Original question and givens

Over R, M=[[1, 1], [0, 23]] is 2×2, λ=23, and the nonzero candidate v=[1, 22]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

• G001 — Over R, M=[[1, 1], [0, 23]] is 2×2, λ=23, and the nonzero candidate v=[1, 22]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

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

Step-by-step derivation

[[1, 1], [0, 23]]

exact real2×2 matrix;no spectrum or diagonalization inferred

row1 product1 := mul(1, 1)

row1 product1 = 1

row1 product2 := mul(1, 22)

row1 product2 = 22

Mv coordinate1 := sum(1, 22)

Mv coordinate1 = 23

λv coordinate1 := mul(23, 1)

λv coordinate1 = 23

residual coordinate1 := sub(23, 23)

residual coordinate1 = 0

row2 product1 := mul(0, 1)

row2 product1 = 0

row2 product2 := mul(23, 22)

row2 product2 = 506

Mv coordinate2 := sum(0, 506)

Mv coordinate2 = 506

λv coordinate2 := mul(23, 22)

λv coordinate2 = 506

residual coordinate2 := sub(506, 506)

residual coordinate2 = 0

[0, 0]

first nonzero coordinate=none

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

none

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.