Lect7 - Chapter 1. Matrices and Systems of Equations...

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Unformatted text preview: Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Finding inverses Method of finding Inverse A I row operations-------------→ I B . Then B is the inverse of A Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Finding inverses Method of finding Inverse A I row operations-------------→ I B . Then B is the inverse of A Explanation: Suppose A I is reduced to I B via some elementary row operations. Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Finding inverses Method of finding Inverse A I row operations-------------→ I B . Then B is the inverse of A Explanation: Suppose A I is reduced to I B via some elementary row operations. Hence, there are elementary matrices such that E k ··· E 1 ( A I ) = ( I B ) . i.e. E k ··· E 1 A = I and E k ··· E 1 I = B . Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Finding inverses Method of finding Inverse A I row operations-------------→ I B . Then B is the inverse of A Explanation: Suppose A I is reduced to I B via some elementary row operations. Hence, there are elementary matrices such that E k ··· E 1 ( A I ) = ( I B ) . i.e. E k ··· E 1 A = I and E k ··· E 1 I = B . This implies B = E k ··· E 1 is invertible, and BA = I . i.e. A = B- 1 . ∴ A is invertible and A- 1 = B . (See §3 Ex. 16.) Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Finding inverses (Cont’d) Example . Compute A- 1 if A = 1 4 3- 1- 2 2 2 3 . Express A- 1 as a product of elementary matrices. Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Finding inverses (Cont’d) Ans . Refer to p.66 of textbook: 1 4 3- 1- 2 2 2 3 1 1 1 -------------→ 1 1 1- 1 2- 1 2 1 2 1 4- 1 4- 1 4 1 6 1 2 1 6 via the following eight elementary row operations: R 1 + R 2- 2 R 1 + R 3 3 R 2 + R 3- 1 2 R 3 + R 1- 1 2 R 3 + R 2- 2 R 2 + R 1 1 2 R 2 1 6 R 3 Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Finding inverses (Cont’d) Ans . Refer to p.66 of textbook: 1 4 3- 1- 2 2 2 3...
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Lect7 - Chapter 1. Matrices and Systems of Equations...

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