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 Proof of Theorem (Cont’d) Type II: Multiply the i th row by α ( 6 = ) . E = I i- 1 α I n- i where α 6 = . Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Proof of Theorem (Cont’d) Type II: Multiply the i th row by α ( 6 = ) . E = I i- 1 α I n- i where α 6 = . By direct checking, E- 1 = I i- 1 1 α I n- i Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Proof of Theorem (Cont’d) Type III: Add mR i to R j E = I 1 . . . m 1 I i th row j th row = I + mZ ji where Z ji = 1 j th i th . Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Proof of Theorem (Cont’d) Type III: Add mR i to R j E = I 1 . . . m 1 I i th row j th row = I + mZ ji where Z ji = 1 j th i th . Then, E- 1 = I- mZ ji . Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Row Equivalent Definition B is row equivalent to A if there exist elementary matrices E 1 , E 2 , ··· , E k such that B = E k E k- 1 ··· E 1 A . Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Row Equivalent Definition B is row equivalent to A if there exist elementary matrices E 1 , E 2 , ··· , E k such that B = E k E k- 1 ··· E 1 A . Remark In other words, B can be obtained by a finite number of row operations. Chapter 1. Matrices and Systems of Equations Math1111 Elementary Matrices Row Equivalent Definition B is row equivalent to A if there exist elementary matrices E 1 , E 2 , ··· , E k such that B = E k E k- 1 ··· E 1 A . Remark In other words, B can be obtained by a finite number of row operations. Nice properties 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 ) ....
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Lect7 - Chapter 1. Matrices and Systems of Equations...

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