Question 31483: Kernel Membership for a Candidate Scalar

Original question and givens

Over R, the 2×2 matrix is M=[[-5, 0], [0, 9]]. For the candidate scalar λ=-4 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=[[-5, 0], [0, 9]]. For the candidate scalar λ=-4 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

Step-by-step derivation

source roles = [qf-eigenvalue-kernel-membership, kernel_membership, domain, reals]

derived = [[equality_including_zero_vector_YesNo]]

Eigen ORIGINAL row0 term0 := mul(-5, 1)

Eigen ORIGINAL row0 term0 = -5

Eigen ORIGINAL row0 term1 := mul(0, -1)

Eigen ORIGINAL row0 term1 = 0

Eigen ORIGINAL Mv row0 sum := sum(-5, 0)

Eigen ORIGINAL Mv row0 sum = -5

Eigen ORIGINAL lambda times v row0 := mul(-4, 1)

Eigen ORIGINAL lambda times v row0 = -4

Eigen ORIGINAL residual row0 := sub(-5, -4)

Eigen ORIGINAL residual row0 = -1

Eigen ORIGINAL row1 term0 := mul(0, 1)

Eigen ORIGINAL row1 term0 = 0

Eigen ORIGINAL row1 term1 := mul(9, -1)

Eigen ORIGINAL row1 term1 = -9

Eigen ORIGINAL Mv row1 sum := sum(0, -9)

Eigen ORIGINAL Mv row1 sum = -9

Eigen ORIGINAL lambda times v row1 := mul(-4, -1)

Eigen ORIGINAL lambda times v row1 = 4

Eigen ORIGINAL residual row1 := sub(-9, 4)

Eigen ORIGINAL residual row1 = -13

source roles = [[[-5, 0], [0, 9]], -4, [1, -1], kernel_membership]

derived = [[-5, -9], [-4, 4], [-1, -13], No, Yes, No]

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.