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# Lecture_5 - Lecture Note 5 Dr Jeff Chak-Fu WONG Department...

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Lecture Note 5 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1

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Elementary Matrices Produced by Jeff Chak-Fu 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 non-zero 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. 0 1 1 0 1 0 0 - 2 1 0 0 0 1 2 0 0 0 1 0 0 0 1 Produced by Jeff Chak-Fu WONG 3

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What happens when a matrix is multiplied (on the left) by an elementary matrix? The next examples give a clue. 0 1 1 0 2 3 1 2 = 1 2 2 3 1 0 0 - 2 1 0 0 0 1 1 2 3 2 3 5 - 1 4 7 = 1 2 3 0 - 1 - 1 - 1 4 7 2 0 0 0 1 0 0 0 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 Chak-Fu WONG 4
Type I An elementary matrix of type I is a matrix obtained by interchanging two rows of I n . Produced by Jeff Chak-Fu WONG 5

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Example 1 E 1 = 0 1 0 1 0 0 0 0 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 = 0 1 0 1 0 0 0 0 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 0 1 0 1 0 0 0 0 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 of A . Right multiplication of A by E 1 is equivalent to the elementary column operation of interchanging the first and second columns .
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Lecture_5 - Lecture Note 5 Dr Jeff Chak-Fu WONG Department...

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