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R0006-03738 · q-20261007-advance-20261009-r07-eigenvalue-0038
Over R, M=[[12, 1], [12, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).
G001 — Over R, M=[[12, 1], [12, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).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
[[12, 1], [12, 0]]
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(12, 0)
After
trace = 12
Conditions: none
The trace is the sum of diagonal entries.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
diagonal product := mul(12, 0)
After
diagonal product = 0
Conditions: none
Compute the determinant term a·d.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
off-diagonal product := mul(1, 12)
After
off-diagonal product = 12
Conditions: none
Compute b·c,which must be subtracted rather than added.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
determinant := sub(0, 12)
After
determinant = -12
Conditions: none
Compute ad−bc exactly.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
negative trace := mul(-1, 12)
After
negative trace = -12
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−12)(t−0)−(1)(12)
After
t²+(-12)t+(-12)
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
Before
trace squared := square(12)
After
trace squared = 144
Conditions: none
The quadratic discriminant uses the squared linear coefficient,so the sign vanishes.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
four times determinant := mul(4, -12)
After
four times determinant = -48
Conditions: none
Compute4det completely;the determinant may be negative.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
characteristic discriminant := sub(144, -48)
After
characteristic discriminant = 192
Conditions: none
Δ=trace²−4det;do not take a floating square root or invent real roots.
Rule: exact_rational_arithmetic · Uses: G001, S009
S010 · premise_and_result_audit — Recompute characteristic discriminant with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute four times determinant with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute trace squared with exact rational arithmetic, not rounded decimals.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.192
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: 5d022309e401f5bf1911ead598a492d146d86f290e8062f24cfab5a5cb368557candidate record SHA-256: 677fa9b82cb769342abd106efa83daff601313c74f64c870b139b496d425190dsemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED