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Lect3 - Chapter 1 Matrices and Systems of Equations...

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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 2 Example . Solve the system x 1 + x 2 + x 3 + x 4 + x 5 = 1 - x 1 - x 2 + x 5 = - 1 - 2 x 1 - 2 x 2 + 3 x 5 = 1 x 3 + x 4 + 3 x 5 = 3 x 1 + x 2 + 2 x 3 + 2 x 4 + 4 x 5 = 4 if its augmented matrix is reduced to 1 1 1 1 1 0 0 1 1 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 3 0 0
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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Lead & Free variables Lead variables & Free variables A leading one corresponds to the presence of a variable, called a lead variable All remaining variables are called free variables
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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Lead & Free variables Lead variables & Free variables A leading one corresponds to the presence of a variable, called a lead variable All remaining variables are called free variables Remark A system has infinitely many solutions if and only if it has free variable(s).
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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 3 Example . The augmented matrix of the system x 1 + x 2 + x 3 + x 4 + x 5 = 1 - x 1 - x 2 + x 5 = - 1 - 2 x 1 - 2 x 2 + 3 x 5 = 1 x 3 + x 4 + 3 x 5 = - 1 x 1 + x 2 + 2 x 3 + 2 x 4 + 4 x 5 = 1 is reduced to 1 1 1 1 1 0 0 1 1 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 3 - 4 - 3 Solve the system.
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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 3 Example . The augmented matrix of the system x 1 + x 2 + x 3 + x 4 + x 5 = 1 - x 1 - x 2 + x 5 = - 1 - 2 x 1 - 2 x 2 + 3 x 5 = 1 x 3 + x 4 + 3 x 5 = - 1 x 1 + x 2 + 2 x 3 + 2 x 4 + 4 x 5 = 1 is reduced to 1 1 1 1 1 0 0 1 1 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 3 - 4 - 3 Solve the system. Consistent system A linear system is consistent if it has solution(s); otherwise, it is inconsistent .
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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form Row Echelon Form (i) The first entry in each nonzero row is 1. (ii) The number of leading zeros of nonzero rows increases from the top to the bottom. (iii) All zero rows, if any, are at the bottom.
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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form Row Echelon Form (i) The first entry in each nonzero row is 1. (ii) The number of leading zeros of nonzero rows increases from the top to the bottom. (iii) All zero rows, if any, are at the bottom. Examples: 1 4 2 0 1 3 0 0 1 , 1 2 3 0 0 1 0 0 0 , 1 3 1 0 0 0 1 3 0 0 0 0
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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form Row Echelon Form (i) The first entry in each nonzero row is 1.
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