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(a) Let C₁, C2, C3, and ♬ be the columns given below. Let C be the matrix C = [ ₁ ₂ 3 ]. That is, C is the matrix with columns C₁, C2, and c3, in that order. = 1 2 " = 2 C3 = 0 > a -8 S = b Compute the product Cs, and show that it is equal to the linear combination of c₁, c2, and 3 with coefficients a, b, and c. (b) Let d = 7c₁ +872 +973. Show that the system C = d is consistent. (Hint: Use the result of part (a). You don't have to row reduce anything here!) (c) Conclude from the ideas in the previous two parts that the system of equations Cx = has a solution if and only if b is equal to a linear combination of 1, 2, 3 (the columns of C).

(a) Let C₁, C2, C3, and ♬ be the columns given below. Let C be the matrix C = [ ₁ ₂ 3 ]. That is, C is the matrix with columns C₁, C2, and c3, in that order. = 1 2 " = 2 C3 = 0 > a -8 S = b Compute the product Cs, and show that it is equal to the linear combination of c₁, c2, and 3 with coefficients a, b, and c. (b) Let d = 7c₁ +872 +973. Show that the system C = d is consistent. (Hint: Use the result of part (a). You don't have to row reduce anything here!) (c) Conclude from the ideas in the previous two parts that the system of equations Cx = has a solution if and only if b is equal to a linear combination of 1, 2, 3 (the columns of C).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 30E
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(a) Let č₁, C2, C3, and s' be the columns given below. Let C be the matrix C = [ ₁ ₂ 3 ]. That is, C is the matrix with columns č₁, №2, and №3, in that order. --8-8-8-8 2 2 3 0 = = a b Compute the product Cs, and show that it is equal to the linear combination of c₁, c2, and 3 with coefficients a, b, and c. (b) Let đ= 7c₁ +872 +973. Show that the system C = d is consistent. (Hint: Use the result of part (a). You don't have to row reduce anything here!) (c) Conclude from the ideas in the previous two parts that the system of equations Cỡ = b has a solution if and only if b is equal to a linear combination of c₁, C2, C3 (the columns of C).
Transcribed Image Text:(a) Let č₁, C2, C3, and s' be the columns given below. Let C be the matrix C = [ ₁ ₂ 3 ]. That is, C is the matrix with columns č₁, №2, and №3, in that order. --8-8-8-8 2 2 3 0 = = a b Compute the product Cs, and show that it is equal to the linear combination of c₁, c2, and 3 with coefficients a, b, and c. (b) Let đ= 7c₁ +872 +973. Show that the system C = d is consistent. (Hint: Use the result of part (a). You don't have to row reduce anything here!) (c) Conclude from the ideas in the previous two parts that the system of equations Cỡ = b has a solution if and only if b is equal to a linear combination of c₁, C2, C3 (the columns of C).
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