Question 31269: Determinant of a Matrix Product

Original question and givens

Over R, A=[[6, 3], [-2, 19]] and B=[[19, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.

• G001 — Over R, A=[[6, 3], [-2, 19]] and B=[[19, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.

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

source roles = [qf-determinant-product, product_determinant, domain, reals]

derived = [[integer_det_AB]]

Det AB row0 column0 term0 := mul(6, 19)

Det AB row0 column0 term0 = 114

Det AB row0 column0 term1 := mul(3, 0)

Det AB row0 column0 term1 = 0

Det AB row0 column0 sum := sum(114, 0)

Det AB row0 column0 sum = 114

Det AB row0 column1 term0 := mul(6, -1)

Det AB row0 column1 term0 = -6

Det AB row0 column1 term1 := mul(3, 2)

Det AB row0 column1 term1 = 6

Det AB row0 column1 sum := sum(-6, 6)

Det AB row0 column1 sum = 0

Det AB row1 column0 term0 := mul(-2, 19)

Det AB row1 column0 term0 = -38

Det AB row1 column0 term1 := mul(19, 0)

Det AB row1 column0 term1 = 0

Det AB row1 column0 sum := sum(-38, 0)

Det AB row1 column0 sum = -38

Det AB row1 column1 term0 := mul(-2, -1)

Det AB row1 column1 term0 = 2

Det AB row1 column1 term1 := mul(19, 2)

Det AB row1 column1 term1 = 38

Det AB row1 column1 sum := sum(2, 38)

Det AB row1 column1 sum = 40

source roles = [[[6, 3], [-2, 19]], [[19, -1], [0, 2]]]

derived = [[[114, 0], [-38, 40]]]

Det requested matrix ORIGINAL main diagonal product := mul(114, 40)

Det requested matrix ORIGINAL main diagonal product = 4560

Det requested matrix ORIGINAL other diagonal product := mul(0, -38)

Det requested matrix ORIGINAL other diagonal product = 0

Det requested matrix ORIGINAL signed ad minus bc := sub(4560, 0)

Det requested matrix ORIGINAL signed ad minus bc = 4560

source roles = [Det requested matrix, [[114, 0], [-38, 40]]]

derived = [4560, 0, 4560]

Reverse checks

• S018 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S017 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL signed ad minus bc with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL other diagonal product with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Det requested matrix ORIGINAL main diagonal product with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S013 · premise_and_result_audit — Recompute Det AB row1 column1 sum with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Det AB row1 column1 term1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Det AB row1 column1 term0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Det AB row1 column0 sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Det AB row1 column0 term1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Det AB row1 column0 term0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Det AB row0 column1 sum with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Det AB row0 column1 term1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Det AB row0 column1 term0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Det AB row0 column0 sum with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Det AB row0 column0 term1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Det AB row0 column0 term0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.

Result

4560

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.