Question 32856: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-7, 6] and c=-1, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-7, 6] 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, -7)

Homogeneity ORIGINAL c times u ORIGINAL component0 = 7

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(-1, 6)

Homogeneity ORIGINAL c times u ORIGINAL component1 = -6

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(-7, 7)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = -49

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(2, -6)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = -12

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(-49, -12)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = -61

source roles = [Homogeneity LEFT M of scaled input, 0, [-7, 2], [7, -6]]

derived = [[-49, -12], -61]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, 7)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = -14

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(6, -6)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = -36

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(-14, -36)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = -50

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 6], [7, -6]]

derived = [[-14, -36], -50]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(-7, -7)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 49

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(2, 6)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 12

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(49, 12)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 61

source roles = [Homogeneity RIGHT M of u, 0, [-7, 2], [-7, 6]]

derived = [[49, 12], 61]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, -7)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = 14

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(6, 6)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 36

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(14, 36)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 50

source roles = [Homogeneity RIGHT M of u, 1, [-2, 6], [-7, 6]]

derived = [[14, 36], 50]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(-1, 61)

Homogeneity RIGHT c times image ORIGINAL component0 = -61

Homogeneity RIGHT c times image ORIGINAL component1 := mul(-1, 50)

Homogeneity RIGHT c times image ORIGINAL component1 = -50

source roles = [homogeneity_values, [[-7, 2], [-2, 6]], [-7, 6], -1]

derived = [[-61, -50], [-61, -50], [-61, -50] = [-61, -50]]

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

[-61, -50] = [-61, -50]

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.