Question 32874: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[11, 24] and c=-4, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[11, 24] and c=-4, compute T(cu) and cT(u). Answer as left vector = right vector.

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-linear-transformation

Step-by-step derivation

source roles = [qf-linear-map-homogeneity-values, homogeneity_values, reals, []]

derived = [[VECTOR_EQUALS_VECTOR_STRING]]

Homogeneity ORIGINAL c times u ORIGINAL component0 := mul(-4, 11)

Homogeneity ORIGINAL c times u ORIGINAL component0 = -44

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(-4, 24)

Homogeneity ORIGINAL c times u ORIGINAL component1 = -96

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(11, -44)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = -484

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(4, -96)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = -384

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(-484, -384)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = -868

source roles = [Homogeneity LEFT M of scaled input, 0, [11, 4], [-44, -96]]

derived = [[-484, -384], -868]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -44)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 88

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(24, -96)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = -2304

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(88, -2304)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = -2216

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 24], [-44, -96]]

derived = [[88, -2304], -2216]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(11, 11)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 121

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(4, 24)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 96

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(121, 96)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 217

source roles = [Homogeneity RIGHT M of u, 0, [11, 4], [11, 24]]

derived = [[121, 96], 217]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 11)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = -22

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(24, 24)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 576

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(-22, 576)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 554

source roles = [Homogeneity RIGHT M of u, 1, [-2, 24], [11, 24]]

derived = [[-22, 576], 554]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(-4, 217)

Homogeneity RIGHT c times image ORIGINAL component0 = -868

Homogeneity RIGHT c times image ORIGINAL component1 := mul(-4, 554)

Homogeneity RIGHT c times image ORIGINAL component1 = -2216

source roles = [homogeneity_values, [[11, 4], [-2, 24]], [11, 24], -4]

derived = [[-868, -2216], [-868, -2216], [-868, -2216] = [-868, -2216]]

Reverse checks

• S022 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S021 · premise_and_result_audit — Recompute Homogeneity RIGHT c times image ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Homogeneity RIGHT c times image ORIGINAL component0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S018 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S014 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S010 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S006 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Homogeneity ORIGINAL c times u ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Homogeneity ORIGINAL c times u ORIGINAL component0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

[-868, -2216] = [-868, -2216]

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.