Question 3781: Ordered Residual of a Specified Kernel Equation

Original question and givens

Over R, M=[[6, 3], [-1, -3]] is 2×2, λ=-2, and v=[3, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

• G001 — Over R, M=[[6, 3], [-1, -3]] is 2×2, λ=-2, and v=[3, 2]. 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

Step-by-step derivation

[[6, 3], [-1, -3]]

exact real2×2 matrix;no spectrum or diagonalization inferred

row1 product1 := mul(6, 3)

row1 product1 = 18

row1 product2 := mul(3, 2)

row1 product2 = 6

Mv coordinate1 := sum(18, 6)

Mv coordinate1 = 24

λv coordinate1 := mul(-2, 3)

λv coordinate1 = -6

residual coordinate1 := sub(24, -6)

residual coordinate1 = 30

row2 product1 := mul(-1, 3)

row2 product1 = -3

row2 product2 := mul(-3, 2)

row2 product2 = -6

Mv coordinate2 := sum(-3, -6)

Mv coordinate2 = -9

λv coordinate2 := mul(-2, 2)

λv coordinate2 = -4

residual coordinate2 := sub(-9, -4)

residual coordinate2 = -5

v=[3, 2];residual=(30, -5)

kernel equation is not an eigenvector certificate for v=0

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

(30, -5)

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.