# Lect6 - Chapter 1. Matrices and Systems of Equations...

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Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) Remark Elementary row operations preserve equivalent linear systems. Recall: Two linear systems are equivalent if they have the same solutions.

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Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) Remark Elementary row operations preserve equivalent linear systems. Question hinted When do A x = b and MA x = M b have the same solutions for x ?
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) Remark Elementary row operations preserve equivalent linear systems. Question hinted When do A x = b and MA x = M b have the same solutions for x ? Observation: ( * ) A x = b ( ** ) MA x = M b i.e. Solution of Equation ( * ) is a solution of Equation ( ** ) .

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Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) Remark Elementary row operations preserve equivalent linear systems. Question hinted When do A x = b and MA x = M b have the same solutions for x ? Observation: ( * ) A x = b ( ** ) MA x = M b i.e. Solution of Equation ( * ) is a solution of Equation ( ** ) . Question Is the converse true? (i.e. Is a solution of ( ** ) a solution of ( * ) ?)
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) Remark Elementary row operations preserve equivalent linear systems. Question hinted When do A x = b and MA x = M b have the same solutions for x ? Observation: ( * ) A x = b ( ** ) MA x = M b i.e. Solution of Equation ( * ) is a solution of Equation ( ** ) . Question Is the converse true? (i.e. Is a solution of ( ** ) a solution of ( * ) ?) If M is invertible , the answer is YES . i.e. The action of multiplying invertible matrices preserves the solutions of matrix equations.

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Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) a 11 x 1 + ··· + a 1 n x n = b 1 . . . a m 1 x 1 + + a mn x n = b m ←-----------→ A x = b elementary row operations ? a 0 11 x 1 + + a 0 1 n x n = b 0 1 . . . a 0 m 1 x 1 + + a 0 mn x n = b 0 m A 0 x = b 0
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) a 11 x 1 + ··· + a 1 n x n = b 1 . . . a m 1 x 1 + + a mn x n = b m ←-----------→ A x = b elementary row operations ? a 0 11 x 1 + + a 0 1 n x n = b 0 1 . . . a 0 m 1 x 1 + + a 0 mn x n = b 0 m A 0 x = b 0 Guess A 0 = MA , b 0 = M b for some invertible M

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Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) a 11 x 1 + ··· + a 1 n x n = b 1 . . . a m 1 x 1 + + a mn x n = b m ←-----------→ A x = b elementary row operations ? a 0 11 x 1 + + a 0 1 n x n = b 0 1 . . . a 0 m 1 x 1 + + a 0 mn x n = b 0 m A 0 x = b 0 Guess A 0 = MA , b 0 = M b for some invertible M M = ?
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Equivalent Systems (Cont’d) a 11 x 1 + ··· + a 1 n x n = b 1 .

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## This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.

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Lect6 - Chapter 1. Matrices and Systems of Equations...

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