math125-lecture 3.2 - A matrix B is the multiplicative...

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Unformatted text preview: A matrix B is the multiplicative inverse of the matrix A if A B = I and B A = I Notation: The multiplicative inverse of A is denoted A-1 For a real number a , if b satisfies ab = 1 = ba , then b = = a-1 1 a Sec 3.2 Saturday, February 06, 2010 11:16 PM Section3dot2 Page 1 A matrix B is the multiplicative inverse of the matrix A if A B = I and B A = I Notation: The multiplicative inverse of A is denoted A-1 REMARKS: 1. If there is an inverse, there is just one 2. Only square matrices have inverses (why? check dimensions !) 3. Not every square matrix has an inverse For a real number a , if b satisfies ab = 1 = ba , then b = = a-1 1 a 3.2.1 Saturday, February 06, 2010 11:16 PM Section3dot2 Page 2 A matrix B is the multiplicative inverse of the matrix A if A B = I and B A = I Notation: The multiplicative inverse of A is denoted A-1 REMARKS: 1. If there is an inverse, there is just one 2. Only square matrices have inverses (why? check dimensions !) 3. Not every square matrix has an inverse For a real number a , if b satisfies ab = 1 = ba , then b = = a-1 1 a If A is invertible ( i.e. has an inverse), then A can be changed through row operations into the identity matrix rref ( If A is invertible, then A I ) If the inverse B is written in terms of its columns B = [ b 1 b 2 . . . b n ], then A B = A [ b 1 b 2 . . . b n ] = [ A b 1 A b 2 . . . A b n ] = I = [ e 1 e 2 . . . e n ] So A b 1 = e 1 , A b 2 = e 2 , . . ., A b n = e n a multisystem ! ! (sec 2.2) 3.2.2 Saturday, February 06, 2010 11:16 PM Section3dot2 Page 3 A matrix B is the multiplicative inverse of the matrix A if A B = I and B A = I Notation: The multiplicative inverse of A is denoted A-1 REMARKS: 1. If there is an inverse, there is just one 2. Only square matrices have inverses (why? check dimensions !) 3. Not every square matrix has an inverse For a real number a , if b satisfies ab = 1 = ba , then b = = a-1 1 a If A is invertible ( i.e. has an inverse), then A can be changed through row operations into the identity matrix rref ( If A is invertible, then A I ) If the inverse B is written in terms of its columns B = [ b 1 b 2 . . . b n ], then A B = A [ b 1 b 2 . . . b n ] = [ A b 1 A b 2 . . . A b n ] = I = [ e 1 e 2 . . . e n ] So A b 1 = e 1 , A b 2 = e 2 , . . ., A b n = e n a multisystem ! ! (sec 2.2) Form the augmented matrix [ A | e 1 e 2 . . . e n ] = [ A | I ] and then put [ A | I ] into reduced row echelon form The rref form is [ I | b 1 b 2 . . . b n ] where b 1 , b 2 , . . ., b n are the solution vectors of the multi system. But this is just [ I | B ] and B is the inverse of A 3.2.3 Saturday, February 06, 2010 11:16 PM Section3dot2 Page 4 A matrix B is the multiplicative inverse of the matrix A if A B = I and B A = I Notation: The multiplicative inverse of A is denoted A-1 REMARKS: 1. If there is an inverse, there is just one...
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This note was uploaded on 06/07/2011 for the course MATH 125 taught by Professor Tom during the Fall '07 term at University of Illinois at Urbana–Champaign.

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math125-lecture 3.2 - A matrix B is the multiplicative...

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