Question 31255: Determinant of a Matrix Product

Original question and givens

Over R, A=[[-8, 1], [-2, 5]] and B=[[5, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.

• G001 — Over R, A=[[-8, 1], [-2, 5]] and B=[[5, -1], [1, 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(-8, 5)

Det AB row0 column0 term0 = -40

Det AB row0 column0 term1 := mul(1, 1)

Det AB row0 column0 term1 = 1

Det AB row0 column0 sum := sum(-40, 1)

Det AB row0 column0 sum = -39

Det AB row0 column1 term0 := mul(-8, -1)

Det AB row0 column1 term0 = 8

Det AB row0 column1 term1 := mul(1, 2)

Det AB row0 column1 term1 = 2

Det AB row0 column1 sum := sum(8, 2)

Det AB row0 column1 sum = 10

Det AB row1 column0 term0 := mul(-2, 5)

Det AB row1 column0 term0 = -10

Det AB row1 column0 term1 := mul(5, 1)

Det AB row1 column0 term1 = 5

Det AB row1 column0 sum := sum(-10, 5)

Det AB row1 column0 sum = -5

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

Det AB row1 column1 term0 = 2

Det AB row1 column1 term1 := mul(5, 2)

Det AB row1 column1 term1 = 10

Det AB row1 column1 sum := sum(2, 10)

Det AB row1 column1 sum = 12

source roles = [[[-8, 1], [-2, 5]], [[5, -1], [1, 2]]]

derived = [[[-39, 10], [-5, 12]]]

Det requested matrix ORIGINAL main diagonal product := mul(-39, 12)

Det requested matrix ORIGINAL main diagonal product = -468

Det requested matrix ORIGINAL other diagonal product := mul(10, -5)

Det requested matrix ORIGINAL other diagonal product = -50

Det requested matrix ORIGINAL signed ad minus bc := sub(-468, -50)

Det requested matrix ORIGINAL signed ad minus bc = -418

source roles = [Det requested matrix, [[-39, 10], [-5, 12]]]

derived = [-468, -50, -418]

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

-418

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.