This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ) ....
View
Full Document
 Spring '11
 Linear Algebra, Invertible matrix, Nonsingularity

Click to edit the document details