Question 31266: Determinant of a Matrix Product

Original question and givens

Over R, A=[[3, 4], [-2, 16]] and B=[[16, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.

• G001 — Over R, A=[[3, 4], [-2, 16]] and B=[[16, -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(3, 16)

Det AB row0 column0 term0 = 48

Det AB row0 column0 term1 := mul(4, 0)

Det AB row0 column0 term1 = 0

Det AB row0 column0 sum := sum(48, 0)

Det AB row0 column0 sum = 48

Det AB row0 column1 term0 := mul(3, -1)

Det AB row0 column1 term0 = -3

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

Det AB row0 column1 term1 = 8

Det AB row0 column1 sum := sum(-3, 8)

Det AB row0 column1 sum = 5

Det AB row1 column0 term0 := mul(-2, 16)

Det AB row1 column0 term0 = -32

Det AB row1 column0 term1 := mul(16, 0)

Det AB row1 column0 term1 = 0

Det AB row1 column0 sum := sum(-32, 0)

Det AB row1 column0 sum = -32

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

Det AB row1 column1 term0 = 2

Det AB row1 column1 term1 := mul(16, 2)

Det AB row1 column1 term1 = 32

Det AB row1 column1 sum := sum(2, 32)

Det AB row1 column1 sum = 34

source roles = [[[3, 4], [-2, 16]], [[16, -1], [0, 2]]]

derived = [[[48, 5], [-32, 34]]]

Det requested matrix ORIGINAL main diagonal product := mul(48, 34)

Det requested matrix ORIGINAL main diagonal product = 1632

Det requested matrix ORIGINAL other diagonal product := mul(5, -32)

Det requested matrix ORIGINAL other diagonal product = -160

Det requested matrix ORIGINAL signed ad minus bc := sub(1632, -160)

Det requested matrix ORIGINAL signed ad minus bc = 1792

source roles = [Det requested matrix, [[48, 5], [-32, 34]]]

derived = [1632, -160, 1792]

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

1792

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.