# Question 3720: Descending Characteristic Coefficients of a Real 2×2 Matrix

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0006-03720 · q-20261007-advance-20261009-r07-eigenvalue-0020



## Original question and givens

Over R, M=[[7, 2], [-1, 20]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).


- G001 — Over R, M=[[7, 2], [-1, 20]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).





## 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](https://ocme.evemisslab.com/en/mko/mko-eigenvalue/)





## Step-by-step derivation

S001 · impliesBefore

[[7, 2], [-1, 20]]

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.

Step identifiers and dependenciesRule: two_by_two_matrix_scope · Uses: G001





S002 · calculationBefore

trace := sum(7, 20)

After

trace = 27

Conditions: none

The trace is the sum of diagonal entries.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

diagonal product := mul(7, 20)

After

diagonal product = 140

Conditions: none

Compute the determinant term a·d.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

off-diagonal product := mul(2, -1)

After

off-diagonal product = -2

Conditions: none

Compute b·c,which must be subtracted rather than added.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

determinant := sub(140, -2)

After

determinant = 142

Conditions: none

Compute ad−bc exactly.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

negative trace := mul(-1, 27)

After

negative trace = -27

Conditions: none

The linear coefficient of det(tI−M) is−trace,including its sign.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

det(tI−M)=(t−7)(t−20)−(2)(-1)

After

t²+(-27)t+(142)

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.

Step identifiers and dependenciesRule: characteristic_polynomial_expansion · Uses: G001, S006









## Reverse checks


- 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.





## Result

(1, -27, 142)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: e3577995c1506541926b8471b1b6f1af296e991146ec64476948bb36aa394d7c
- candidate record SHA-256: f2360788bcb64a16503c0dfbae6948ba033ae5adfa4462c10021c18facb2136d
- semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
- question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r07-eigenvalue-0020.json)