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R0006-03717 · q-20261007-advance-20261009-r07-eigenvalue-0017
Over R, M=[[4, 2], [1, 17]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).
G001 — Over R, M=[[4, 2], [1, 17]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).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.
Before
[[4, 2], [1, 17]]
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.
Rule: two_by_two_matrix_scope · Uses: G001
Before
trace := sum(4, 17)
After
trace = 21
Conditions: none
The trace is the sum of diagonal entries.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
diagonal product := mul(4, 17)
After
diagonal product = 68
Conditions: none
Compute the determinant term a·d.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
off-diagonal product := mul(2, 1)
After
off-diagonal product = 2
Conditions: none
Compute b·c,which must be subtracted rather than added.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
determinant := sub(68, 2)
After
determinant = 66
Conditions: none
Compute ad−bc exactly.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
negative trace := mul(-1, 21)
After
negative trace = -21
Conditions: none
The linear coefficient of det(tI−M) is−trace,including its sign.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
det(tI−M)=(t−4)(t−17)−(2)(1)
After
t²+(-21)t+(66)
Conditions: none
Expand the2×2 determinant to get descending coefficients(1,−trace,det).Keep this order and the negative trace;the coefficients alone do not establish diagonalizability.
Rule: characteristic_polynomial_expansion · Uses: G001, S006
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute negative trace with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute determinant with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute off-diagonal product with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute diagonal product with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute trace 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.(1, -21, 66)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 27f99390c16e7e271e67bda6448f9a11848ba81df9934e7bfb89dd12fb36c9b6candidate record SHA-256: 0adf41e9b6cf9cb6a46184521411fe68ea1980a14c6ecaa92f78968c184f0303semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED