Question 7316: Signed Determinant versus Absolute-Value Candidate

Original question and givens

Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [2, 1]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).

• G001 — Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [2, 1]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).

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-determinant

Step-by-step derivation

[[3, 1], [2, 1]]

real2×2 integer-entry slice;determinants are signed

M diagonal product := mul(3, 1)

M diagonal product = 3

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

M off-diagonal product = 2

M signed determinant := sub(3, 2)

M signed determinant = 1

candidate minus signed determinant := sub(2, 1)

candidate minus signed determinant = 1

signed D=1;abs(D)=1;candidate=2;defect=1

candidate correct=No;invertible over R=Yes

Reverse checks

• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute candidate minus signed determinant with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute M signed determinant with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute M off-diagonal product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute M diagonal product 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

([[3, 1], [2, 1]], 1, 1, 1, No, Yes)

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.