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1.3 EXERCISES In Exercises 1-4, transform the augmented matrix for the given system to reduced echelon form and, in the notation of Theorem 3, determine n, r, and the number, n-r, of independent variables. If n-r> 0, then identify n -r independent variables. 1. 2x₁ + 2x₂ - xy = -2x₁2x2 + 4xy = 2x₁ + 2x₂ + 5x3 = -2x1-2x2-2x3 = -3 1 1 5

1.3 EXERCISES In Exercises 1-4, transform the augmented matrix for the given system to reduced echelon form and, in the notation of Theorem 3, determine n, r, and the number, n-r, of independent variables. If n-r> 0, then identify n -r independent variables. 1. 2x₁ + 2x₂ - xy = -2x₁2x2 + 4xy = 2x₁ + 2x₂ + 5x3 = -2x1-2x2-2x3 = -3 1 1 5

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 38EQ
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1.3 EXERCISES In Exercises 1-4, transform the augmented matrix for the given system to reduced echelon form and, in the notation of Theorem 3, determine n, r, and the number, n-r, of independent variables. If n -r > 0, then identify n -r independent variables. 1. 2x₁ + 2x₂x3 = 1 -2x₁2x2 + 4x3 = 1 2x₁ + 2x₂ + 5x3 = 5 -2x1- - 2x2 - 2x3 = -3 2. 2x₁ + 2x₂ = 4x1 +5x₂ = 4x₁ + 2x₂ = -2 1 4 3. -X₂ + x3 + x4 = 2 x₁ + 2x₂ + 2x3-X4 = 3 x₂ + 3x₂ + x3 = 2 4. x₂ + 2x₂ + 3x3 + 2x4 = 1 x₁ + 2x2 + 3x3 + 5x4 = 2 2x₁ +4x₂+6x3 + x4 = 1 -X₁-2x2-3x3 + 7x4 = 2 In Exercises 5 and 6, assume that the given system is consistent. For each system determine, in the notation of Theorem 3, all possibilities for the number, r of nonzero rows and the number, n-r, of unconstrained variables. Can the system have a unique solution?
Transcribed Image Text:1.3 EXERCISES In Exercises 1-4, transform the augmented matrix for the given system to reduced echelon form and, in the notation of Theorem 3, determine n, r, and the number, n-r, of independent variables. If n -r > 0, then identify n -r independent variables. 1. 2x₁ + 2x₂x3 = 1 -2x₁2x2 + 4x3 = 1 2x₁ + 2x₂ + 5x3 = 5 -2x1- - 2x2 - 2x3 = -3 2. 2x₁ + 2x₂ = 4x1 +5x₂ = 4x₁ + 2x₂ = -2 1 4 3. -X₂ + x3 + x4 = 2 x₁ + 2x₂ + 2x3-X4 = 3 x₂ + 3x₂ + x3 = 2 4. x₂ + 2x₂ + 3x3 + 2x4 = 1 x₁ + 2x2 + 3x3 + 5x4 = 2 2x₁ +4x₂+6x3 + x4 = 1 -X₁-2x2-3x3 + 7x4 = 2 In Exercises 5 and 6, assume that the given system is consistent. For each system determine, in the notation of Theorem 3, all possibilities for the number, r of nonzero rows and the number, n-r, of unconstrained variables. Can the system have a unique solution?
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