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Lecture 25 - Mar 9

Lecture 25 - Mar 9 - Monday March 9 Lecture 25 Elementary...

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Monday March 9 Lecture 25 : Elementary matrices and invertibility . (Refers to 3.6) Expectations: 1. Define row-equivalent matrices. 2. Recognize that a square matrix A is invertible iff A is row-equivalent to I n . 3. Define elementary matrix. 4. Recognize that for any two row-equivalent matrices A and B , there exist and invertible matrix P such that A = PB . 25.1 Definition Row-equivalent matrices . Two matrices A and B are row-equivalent if there is a finite sequence of elementary row operations that can be applied to A to obtain the matrix B . (Hence row-equivalent matrices both reduce to the same RREF. Why?) We sometimes use the notation A ~ B , to say A is row-equivalent to B . 25.1.1 Example The following chain of ERO's applied to A show shows that A is row-equivalent to B . 0 1 0 1 1 2 1 5 1 2 1 5 1 2 1 5 A = 1 2 1 5 0 1 0 1 0 1 0 1 0 1 0 1 = B 2 2 2 6 P 12 2 2 2 6 (1/2)R 3 1 1 1 3 1R 1 + R 3 0 1 0 2 25.2 Definition An " elementary matrix " is a square matrix E m × m obtained by applying a SINGLE elementary row operation to I m . 25.2.1 There are 3 types of elementary matrices, E Pij , E cR i , E cR i + R j . Each is obtained by applying one of the 3 elementary row operations of type I, II, or II respectively to I m .
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