Question 7376: 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=[[0, 0], [0, 0]], t=0. 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=[[0, 0], [0, 0]], t=0. 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

[[0, 0], [0, 0]]

real2×2 integer-entry slice;determinants are signed

M diagonal product := mul(0, 0)

M diagonal product = 0

M off-diagonal product := mul(0, 0)

M off-diagonal product = 0

M signed determinant := sub(0, 0)

M signed determinant = 0

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

scaled matrix entry0,0 = 0

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

scaled matrix entry0,1 = 0

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

scaled matrix entry1,0 = 0

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

scaled matrix entry1,1 = 0

scaled M diagonal product := mul(0, 0)

scaled M diagonal product = 0

scaled M off-diagonal product := mul(0, 0)

scaled M off-diagonal product = 0

scaled M signed determinant := sub(0, 0)

scaled M signed determinant = 0

matrix scalar squared := square(0)

matrix scalar squared = 0

t squared times original determinant := mul(0, 0)

t squared times original determinant = 0

t=0;D(tM)=0;t²D(M)=0

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

(0, [[0, 0], [0, 0]], 0, 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.