Question 32872: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[9, 22] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[9, 22] and c=-2, 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(-2, 9)

Homogeneity ORIGINAL c times u ORIGINAL component0 = -18

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(-2, 22)

Homogeneity ORIGINAL c times u ORIGINAL component1 = -44

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(9, -18)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = -162

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(2, -44)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = -88

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(-162, -88)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = -250

source roles = [Homogeneity LEFT M of scaled input, 0, [9, 2], [-18, -44]]

derived = [[-162, -88], -250]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -18)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 36

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(22, -44)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = -968

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(36, -968)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = -932

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 22], [-18, -44]]

derived = [[36, -968], -932]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(9, 9)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 81

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(2, 22)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 44

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(81, 44)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 125

source roles = [Homogeneity RIGHT M of u, 0, [9, 2], [9, 22]]

derived = [[81, 44], 125]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 9)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = -18

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(22, 22)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 484

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(-18, 484)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 466

source roles = [Homogeneity RIGHT M of u, 1, [-2, 22], [9, 22]]

derived = [[-18, 484], 466]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(-2, 125)

Homogeneity RIGHT c times image ORIGINAL component0 = -250

Homogeneity RIGHT c times image ORIGINAL component1 := mul(-2, 466)

Homogeneity RIGHT c times image ORIGINAL component1 = -932

source roles = [homogeneity_values, [[9, 2], [-2, 22]], [9, 22], -2]

derived = [[-250, -932], [-250, -932], [-250, -932] = [-250, -932]]

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

[-250, -932] = [-250, -932]

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.