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Lect2

# Lect2 - Chapter 1 Matrices and Systems of Equations...

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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix Observation The variables and equality signs do not take part. For a m × n system a 1 1 x 1 + a 1 2 x 2 + ··· + a 1 n x n = b 1 a 2 1 x 1 + a 2 2 x 2 + ··· + a 2 n x n = b 2 . . . . . . . . . a m 1 x 1 + a m 2 x 2 + ··· + a m n x n = b m the augmented matrix is a rectangular array a 11 a 12 ··· a 1 n b 1 a 21 a 22 ··· a 2 n b 2 . . . . . . . . . . . . . . . a m 1 a m 2 ··· a mn b m The vertical line - separate coefficients & constant terms
Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix - example Example . Write down the augmented matrix of x 1 + 2 x 2 + x 3 = 3 3 x 1 - x 2 - 3 x 3 = - 1 2 x 1 + 3 x 2 + x 3 = 4

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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix - example Example . Write down the augmented matrix of x 1 + 2 x 2 + x 3 = 3 3 x 1 - x 2 - 3 x 3 = - 1 2 x 1 + 3 x 2 + x 3 = 4 Solution . The augmented matrix is 1 2 1 3 3 - 1 - 3 - 1 2 3 1 4
Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Motivation Example . Solve 2 x - y = 9 - x + 7 y = - 10

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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Motivation Example . Solve 2 x - y = 9 - x + 7 y = - 10 Solution . 2 x - y = 9 - x + 7 y = - 10 -------------→ - x + 7 y = - 10 2 x - y = 9 - 2 x + 14 y = - 20 2 x - y = 9 -------------→ - 2 x + 14 y = - 20 13 y = - 11
Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Motivation Example . Solve 2 x - y = 9 - x + 7 y = - 10 Solution . 2 x - y = 9 - x + 7 y = - 10 -------------→ - x + 7 y = - 10 2 x - y = 9 - 2 x + 14 y = - 20 2 x - y = 9 -------------→ - 2 x + 14 y = - 20 13 y = - 11 In terms of augmented matrix, 2 - 1 9 - 1 7 - 10

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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Motivation Example . Solve 2 x - y = 9 - x + 7 y = - 10 Solution . 2 x - y = 9 - x + 7 y = - 10 -------------→ - x + 7 y = - 10 2 x - y = 9 - 2 x + 14 y = - 20 2 x - y = 9 -------------→ - 2 x + 14 y = - 20 13 y = - 11 Interchange the equations In terms of augmented matrix, Interchange rows 2 - 1 9 - 1 7 - 10 -→ - 1 7 - 10 2 - 1 9
Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Motivation Example . Solve 2 x - y = 9 - x + 7 y = - 10 Solution . 2 x - y = 9 - x + 7 y = - 10 -------------→ - x + 7 y = - 10 2 x - y = 9 - 2 x + 14 y = - 20 2 x - y = 9 -------------→ - 2 x + 14 y = - 20 13 y = - 11 Multiply 2 to 1st equation In terms of augmented matrix, Multiply 2 to 1st row 2 - 1 9 - 1 7 - 10 -→ - 1 7 - 10 2 - 1 9 -→ - 2 14 - 20 2 - 1 9

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Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Motivation Example . Solve
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Lect2 - Chapter 1 Matrices and Systems of Equations...

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