Question 32859: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-4, 9] and c=4, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-4, 9] 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, -4)

Homogeneity ORIGINAL c times u ORIGINAL component0 = -16

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(4, 9)

Homogeneity ORIGINAL c times u ORIGINAL component1 = 36

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(-4, -16)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = 64

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(1, 36)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = 36

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(64, 36)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = 100

source roles = [Homogeneity LEFT M of scaled input, 0, [-4, 1], [-16, 36]]

derived = [[64, 36], 100]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -16)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 32

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(9, 36)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = 324

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(32, 324)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = 356

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 9], [-16, 36]]

derived = [[32, 324], 356]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(-4, -4)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 16

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(1, 9)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 9

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(16, 9)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 25

source roles = [Homogeneity RIGHT M of u, 0, [-4, 1], [-4, 9]]

derived = [[16, 9], 25]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, -4)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = 8

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(9, 9)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 81

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(8, 81)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 89

source roles = [Homogeneity RIGHT M of u, 1, [-2, 9], [-4, 9]]

derived = [[8, 81], 89]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(4, 25)

Homogeneity RIGHT c times image ORIGINAL component0 = 100

Homogeneity RIGHT c times image ORIGINAL component1 := mul(4, 89)

Homogeneity RIGHT c times image ORIGINAL component1 = 356

source roles = [homogeneity_values, [[-4, 1], [-2, 9]], [-4, 9], 4]

derived = [[100, 356], [100, 356], [100, 356] = [100, 356]]

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

[100, 356] = [100, 356]

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.