# Question 12741: Evaluate an Ordered Real Bilinear Matrix Formula [Return to catalogue page 64](https://ocme.evemisslab.com/en/questions/page/64/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261027-r04-real-bilinear-form-foundations/#q-20261014-advance-20261027-r04-real-bilinear-form-0016) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0024-12741 · q-20261014-advance-20261027-r04-real-bilinear-form-0016 ## Original question and givens Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. 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=[-2, -2], y=[-1, -3]. Give (matrix,x,y,value). - G001 — Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. 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=[-2, -2], y=[-1, -3]. 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](https://ocme.evemisslab.com/en/mko/mko-real-bilinear-form/) ## Step-by-step derivation S001 · impliesBefore fixed real matrix=[[3, -2], [4, 2]];B(x,y)=sum_i sum_j x_i*A_ij*y_j After distributivity proves real linearity in each slot separately for all vectors and scalars Conditions: none Distributivity expands any real linear combination in either slot of the finite double sum,giving separate universal linearity certificates.Integer instances evaluate this fixed formula,not prove a black box by sampling. Step identifiers and dependenciesRule: matrix_formula_bilinearity_universal_not_samples · Uses: G001 S002 · impliesBefore requested form:matrix=[[3, -2], [4, 2]];first slot=[-2, -2];second slot=[-1, -3] After B(first,second)=first^T*(matrix*second);slot order fixed Conditions: none First multiply A by the second slot,then dot with the first.A general matrix is not symmetric,so the two slots cannot be exchanged. Step identifiers and dependenciesRule: bilinear_ordered_matrix_slots · Uses: G001, S001 S003 · calculationBefore requested form matrix row0 column0 term := mul(3, -1) After requested form matrix row0 column0 term = -3 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore requested form matrix row0 column1 term := mul(-2, -3) After requested form matrix row0 column1 term = 6 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore requested form matrix-times-second row0 := sum(-3, 6) After requested form matrix-times-second row0 = 3 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore requested form matrix row1 column0 term := mul(4, -1) After requested form matrix row1 column0 term = -4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore requested form matrix row1 column1 term := mul(2, -3) After requested form matrix row1 column1 term = -6 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore requested form matrix-times-second row1 := sum(-4, -6) After requested form matrix-times-second row1 = -10 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore requested form first-slot coordinate0 term := mul(-2, 3) After requested form first-slot coordinate0 term = -6 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore requested form first-slot coordinate1 term := mul(-2, -10) After requested form first-slot coordinate1 term = 20 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore requested form final first-slot dot sum := sum(-6, 20) After requested form final first-slot dot sum = 14 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 ## 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 ([[3, -2], [4, 2]], [-2, -2], [-1, -3], 14) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: ce79ce8b4f6656599b767f96024c5542ef73e1fd868f98e1dc18a422f587923a - candidate record SHA-256: b4bce968d302242f438813fd991e6870a36579dac4c7dbb06307e7c25f2eb0c9 - semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35 - question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261027-r04-real-bilinear-form-0016.json)