Question 31262: Determinant of a Matrix Product

Original question and givens

Over R, A=[[-1, 4], [-2, 12]] and B=[[12, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.

• G001 — Over R, A=[[-1, 4], [-2, 12]] and B=[[12, -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(-1, 12)

Det AB row0 column0 term0 = -12

Det AB row0 column0 term1 := mul(4, 2)

Det AB row0 column0 term1 = 8

Det AB row0 column0 sum := sum(-12, 8)

Det AB row0 column0 sum = -4

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

Det AB row0 column1 term0 = 1

Det AB row0 column1 term1 := mul(4, 2)

Det AB row0 column1 term1 = 8

Det AB row0 column1 sum := sum(1, 8)

Det AB row0 column1 sum = 9

Det AB row1 column0 term0 := mul(-2, 12)

Det AB row1 column0 term0 = -24

Det AB row1 column0 term1 := mul(12, 2)

Det AB row1 column0 term1 = 24

Det AB row1 column0 sum := sum(-24, 24)

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(12, 2)

Det AB row1 column1 term1 = 24

Det AB row1 column1 sum := sum(2, 24)

Det AB row1 column1 sum = 26

source roles = [[[-1, 4], [-2, 12]], [[12, -1], [2, 2]]]

derived = [[[-4, 9], [0, 26]]]

Det requested matrix ORIGINAL main diagonal product := mul(-4, 26)

Det requested matrix ORIGINAL main diagonal product = -104

Det requested matrix ORIGINAL other diagonal product := mul(9, 0)

Det requested matrix ORIGINAL other diagonal product = 0

Det requested matrix ORIGINAL signed ad minus bc := sub(-104, 0)

Det requested matrix ORIGINAL signed ad minus bc = -104

source roles = [Det requested matrix, [[-4, 9], [0, 26]]]

derived = [-104, 0, -104]

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

-104

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.