Question 7375: Signed Determinants Before and After a Row Swap

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=[[12, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).

• 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=[[12, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).

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

[[12, 1], [2, 1]]

real2×2 integer-entry slice;determinants are signed

M diagonal product := mul(12, 1)

M diagonal product = 12

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

M off-diagonal product = 2

M signed determinant := sub(12, 2)

M signed determinant = 10

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

row-swapped M diagonal product = 2

row-swapped M off-diagonal product := mul(1, 12)

row-swapped M off-diagonal product = 12

row-swapped M signed determinant := sub(2, 12)

row-swapped M signed determinant = -10

original plus row-swapped determinants := sum(10, -10)

original plus row-swapped determinants = 0

M=[[12, 1], [2, 1]];swapped rows=[[2, 1], [12, 1]]

signed determinants=(10,-10)

Reverse checks

• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute original plus row-swapped determinants with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute row-swapped M signed determinant with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute row-swapped M off-diagonal product with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute row-swapped M diagonal product 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

([[12, 1], [2, 1]], 10, [[2, 1], [12, 1]], -10, 0)

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.