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Unformatted text preview: Lecture Note 5 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff ChakFu WONG 1 Elementary Matrices Produced by Jeff ChakFu WONG 2 We have seen that any matrix can be transformed to row echelon form or reduced row echelon form by means of three elementary row operations: • interchange two rows; • multiply a row by a nonzero scalar; • add a multiple of one row to another row. An n × n matrix is called an elementary matrix if it can be obtained from the identity matrix I n by a single elementary row operation. Here are some examples of elementary matrices. 1 1 1 2 1 1 2 1 1 Produced by Jeff ChakFu WONG 3 What happens when a matrix is multiplied (on the left) by an elementary matrix? The next examples give a clue. • 1 1 2 3 1 2 = 1 2 2 3 • 1 2 1 1 1 2 3 2 3 5 1 4 7 = 1 2 3 1 1 1 4 7 • 2 1 1 1 2 3 4 5 6 7 8 9 = 2 4 6 4 5 6 7 8 9 There are three types of elementary matrices corresponding to the three types of elementary row operations. Produced by Jeff ChakFu WONG 4 Type I An elementary matrix of type I is a matrix obtained by interchanging two rows of I n . Produced by Jeff ChakFu WONG 5 Example 1 E 1 = 1 1 1 E 1 is an elementary matrix of type I, since it was obtained by interchanging the first two rows of I 3 . Let A be a 3 × 3 matrix. E 1 A = 1 1 1 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 21 a 22 a 23 a 11 a 12 a 13 a 31 a 32 a 33 AE 1 = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 1 1 1 = a 12 a 11 a 13 a 22 a 21 a 23 a 32 a 31 a 33 • Multiplying A on the left by E 1 interchanges the first and second rows...
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This note was uploaded on 05/18/2010 for the course MATHEMATIC MAT2310 taught by Professor Dr.jeffchakfuwong during the Spring '06 term at CUHK.
 Spring '06
 Dr.JeffChakFuWONG
 Algebra, Matrices

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