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MATH 2510 Slides 5

# MATH 2510 Slides 5 - Lecture Note 5 Dr Jeff Chak-Fu WONG...

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Lecture Note 5 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong MATH 2510 Linear Algebra and Its Applications Winter, 2011 Produced by Jeff Chak-Fu WONG 1

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E LEMENTARY M ATRICES E LEMENTARY M ATRICES 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 2 0 0 0 1 0 0 0 1 1 0 0 - 2 1 0 0 0 1 E LEMENTARY M ATRICES 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. (Swap) 0 1 1 0 2 3 1 2 = 1 2 2 3 (Scale) 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 (Replacement) 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 There are three types of elementary matrices corresponding to the three types of elementary row operations. E LEMENTARY M ATRICES 4
Type I An elementary matrix of type I is a matrix obtained by interchanging two rows of I n . E LEMENTARY M ATRICES 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 . E LEMENTARY M ATRICES 6
Type II An elementary matrix of type II is a matrix obtained by multiplying a row of I n by a nonzero constant. E LEMENTARY M ATRICES 7

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Example 2 E 2 = 1 0 0 0 1 0 0 0 3 is an elementary matrix of type II. E 2 A = 1 0 0 0 1 0 0 0 3 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 12 a 13 a 21 a 22 a 23 3 a 31 3 a 32 3 a 33 AE 2 = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 1 0 0 0 1 0 0 0 3 = a 11 a 12 3 a 13 a 21 a 22 3 a 23 a 31 a 32 3 a 33 Multiplication on the left by E 2 performs the elementary row operation of multiplying the third row by 3 , while
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MATH 2510 Slides 5 - Lecture Note 5 Dr Jeff Chak-Fu WONG...

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