Question 31271: Determinant of a Matrix Product

Original question and givens

Over R, A=[[8, 1], [-2, 21]] and B=[[21, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.

• G001 — Over R, A=[[8, 1], [-2, 21]] and B=[[21, -1], [2, 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, 21)

Det AB row0 column0 term0 = 168

Det AB row0 column0 term1 := mul(1, 2)

Det AB row0 column0 term1 = 2

Det AB row0 column0 sum := sum(168, 2)

Det AB row0 column0 sum = 170

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 = -6

Det AB row1 column0 term0 := mul(-2, 21)

Det AB row1 column0 term0 = -42

Det AB row1 column0 term1 := mul(21, 2)

Det AB row1 column0 term1 = 42

Det AB row1 column0 sum := sum(-42, 42)

Det AB row1 column0 sum = 0

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

Det AB row1 column1 term0 = 2

Det AB row1 column1 term1 := mul(21, 2)

Det AB row1 column1 term1 = 42

Det AB row1 column1 sum := sum(2, 42)

Det AB row1 column1 sum = 44

source roles = [[[8, 1], [-2, 21]], [[21, -1], [2, 2]]]

derived = [[[170, -6], [0, 44]]]

Det requested matrix ORIGINAL main diagonal product := mul(170, 44)

Det requested matrix ORIGINAL main diagonal product = 7480

Det requested matrix ORIGINAL other diagonal product := mul(-6, 0)

Det requested matrix ORIGINAL other diagonal product = 0

Det requested matrix ORIGINAL signed ad minus bc := sub(7480, 0)

Det requested matrix ORIGINAL signed ad minus bc = 7480

source roles = [Det requested matrix, [[170, -6], [0, 44]]]

derived = [7480, 0, 7480]

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

7480

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.