# Question 7390: Complete Matrix Scaling and Signed Determinant Routes [Return to catalogue page 37](https://ocme.evemisslab.com/en/questions/page/37/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r04-determinant-foundations/#q-20261012-advance-20261016-r04-determinant-0040) 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. R0013-07390 · q-20261012-advance-20261016-r04-determinant-0040 ## 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=[[2, 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=[[2, 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](https://ocme.evemisslab.com/en/mko/mko-determinant/) ## Step-by-step derivation S001 · impliesBefore [[2, 1], [2, 1]] After real2×2 integer-entry slice;determinants are signed Conditions: none Compute the signed determinant ad−bc,not its absolute value;matrix multiplication uses row-by-column,not entrywise products.This finite audit is not a general formal proof. Step identifiers and dependenciesRule: signed_two_by_two_determinant_scope · Uses: G001 S002 · calculationBefore M diagonal product := mul(2, 1) After M diagonal product = 2 Conditions: none Multiply the main diagonal entries a·d. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore M off-diagonal product := mul(1, 2) After M off-diagonal product = 2 Conditions: none Multiply the off-diagonal entries b·c separately,retaining subtraction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore M signed determinant := sub(2, 2) After M signed determinant = 0 Conditions: none Subtract the terms,retaining the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore scaled matrix entry0,0 := mul(1, 2) After scaled matrix entry0,0 = 2 Conditions: none Multiply every matrix cell by the same t. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore scaled matrix entry0,1 := mul(1, 1) After scaled matrix entry0,1 = 1 Conditions: none Multiply every matrix cell by the same t. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore scaled matrix entry1,0 := mul(1, 2) After scaled matrix entry1,0 = 2 Conditions: none Multiply every matrix cell by the same t. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore scaled matrix entry1,1 := mul(1, 1) After scaled matrix entry1,1 = 1 Conditions: none Multiply every matrix cell by the same t. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore scaled M diagonal product := mul(2, 1) After scaled M diagonal product = 2 Conditions: none Multiply the main diagonal entries a·d. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore scaled M off-diagonal product := mul(1, 2) After scaled M off-diagonal product = 2 Conditions: none Multiply the off-diagonal entries b·c separately,retaining subtraction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore scaled M signed determinant := sub(2, 2) After scaled M signed determinant = 0 Conditions: none Subtract the terms,retaining the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore matrix scalar squared := square(1) After matrix scalar squared = 1 Conditions: none Each of the two rows contributes a scale factor,so the2×2 determinant uses t². Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore t squared times original determinant := mul(1, 0) After t squared times original determinant = 0 Conditions: none Separately compute t²D(M) and compare with direct D(tM). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore t=1;D(tM)=0;t²D(M)=0 After whole-matrix2×2 scaling,including negative and zero t Conditions: none Scaling the whole2×2 matrix is not scaling just one row,so the determinant factor is not merely t.A negative t squares positively,and t=0 gives the zero matrix. Step identifiers and dependenciesRule: two_by_two_whole_matrix_scale · Uses: G001, S013 ## 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, [[2, 1], [2, 1]], 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 84f44f2f6ab60d01c0ec98e11b43b2aafa428c682d82fbcbaf2866ee0e5a4594 - candidate record SHA-256: b5aca09bc68f79755baf559ead6965075e71fd9400a74cc34bffbe519e1a4b73 - semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59 - question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r04-determinant-0040.json)