Question 7398: Complete Matrix Scaling and Signed Determinant Routes

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=[[10, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.

• 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=[[10, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.

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

[[10, 1], [2, 1]]

real2×2 integer-entry slice;determinants are signed

M diagonal product := mul(10, 1)

M diagonal product = 10

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

M off-diagonal product = 2

M signed determinant := sub(10, 2)

M signed determinant = 8

scaled matrix entry0,0 := mul(1, 10)

scaled matrix entry0,0 = 10

scaled matrix entry0,1 := mul(1, 1)

scaled matrix entry0,1 = 1

scaled matrix entry1,0 := mul(1, 2)

scaled matrix entry1,0 = 2

scaled matrix entry1,1 := mul(1, 1)

scaled matrix entry1,1 = 1

scaled M diagonal product := mul(10, 1)

scaled M diagonal product = 10

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

scaled M off-diagonal product = 2

scaled M signed determinant := sub(10, 2)

scaled M signed determinant = 8

matrix scalar squared := square(1)

matrix scalar squared = 1

t squared times original determinant := mul(1, 8)

t squared times original determinant = 8

t=1;D(tM)=8;t²D(M)=8

whole-matrix2×2 scaling,including negative and zero t

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute t squared times original determinant with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute matrix scalar squared with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute scaled M signed determinant with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute scaled M off-diagonal product with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute scaled M diagonal product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute scaled matrix entry1,1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute scaled matrix entry1,0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute scaled matrix entry0,1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute scaled matrix entry0,0 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

(8, [[10, 1], [2, 1]], 8, 8)

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.