Question 12734: Evaluate an Ordered Real Bilinear Matrix Formula

Original question and givens

Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value).

• G001 — Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value).

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-real-bilinear-form

Step-by-step derivation

fixed real matrix=[[-4, -2], [2, 4]];B(x,y)=sum_i sum_j x_i*A_ij*y_j

distributivity proves real linearity in each slot separately for all vectors and scalars

requested form:matrix=[[-4, -2], [2, 4]];first slot=[1, -2];second slot=[3, 4]

B(first,second)=first^T*(matrix*second);slot order fixed

requested form matrix row0 column0 term := mul(-4, 3)

requested form matrix row0 column0 term = -12

requested form matrix row0 column1 term := mul(-2, 4)

requested form matrix row0 column1 term = -8

requested form matrix-times-second row0 := sum(-12, -8)

requested form matrix-times-second row0 = -20

requested form matrix row1 column0 term := mul(2, 3)

requested form matrix row1 column0 term = 6

requested form matrix row1 column1 term := mul(4, 4)

requested form matrix row1 column1 term = 16

requested form matrix-times-second row1 := sum(6, 16)

requested form matrix-times-second row1 = 22

requested form first-slot coordinate0 term := mul(1, -20)

requested form first-slot coordinate0 term = -20

requested form first-slot coordinate1 term := mul(-2, 22)

requested form first-slot coordinate1 term = -44

requested form final first-slot dot sum := sum(-20, -44)

requested form final first-slot dot sum = -64

Reverse checks

• S011 · premise_and_result_audit — Recompute requested form final first-slot dot sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute requested form first-slot coordinate1 term with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute requested form first-slot coordinate0 term with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute requested form matrix-times-second row1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute requested form matrix row1 column1 term with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute requested form matrix row1 column0 term with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute requested form matrix-times-second row0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute requested form matrix row0 column1 term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute requested form matrix row0 column0 term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[-4, -2], [2, 4]], [1, -2], [3, 4], -64)

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.