Question 32866: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[3, 16] and c=-1, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[3, 16] and c=-1, 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(-1, 3)

Homogeneity ORIGINAL c times u ORIGINAL component0 = -3

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(-1, 16)

Homogeneity ORIGINAL c times u ORIGINAL component1 = -16

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(3, -3)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = -9

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(4, -16)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = -64

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(-9, -64)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = -73

source roles = [Homogeneity LEFT M of scaled input, 0, [3, 4], [-3, -16]]

derived = [[-9, -64], -73]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -3)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 6

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(16, -16)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = -256

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(6, -256)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = -250

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 16], [-3, -16]]

derived = [[6, -256], -250]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(3, 3)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 9

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(4, 16)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 64

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(9, 64)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 73

source roles = [Homogeneity RIGHT M of u, 0, [3, 4], [3, 16]]

derived = [[9, 64], 73]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 3)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = -6

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(16, 16)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 256

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(-6, 256)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 250

source roles = [Homogeneity RIGHT M of u, 1, [-2, 16], [3, 16]]

derived = [[-6, 256], 250]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(-1, 73)

Homogeneity RIGHT c times image ORIGINAL component0 = -73

Homogeneity RIGHT c times image ORIGINAL component1 := mul(-1, 250)

Homogeneity RIGHT c times image ORIGINAL component1 = -250

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

derived = [[-73, -250], [-73, -250], [-73, -250] = [-73, -250]]

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

[-73, -250] = [-73, -250]

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.