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# section16 - Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1...

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Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1 Lecture Notes Fall 2007 Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Chapter 1 1.6 Elementary Matrices Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla

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Chapter 1 1.6 Elementary Matrices What Is An Elementary Matrix? An elementary matrix E is a matrix obtained from I by using a single elementary row operation. Here are three 3 × 3 examples, illustrating each type of elementary row operation: I = 1 0 0 0 1 0 0 0 1 R 2 R 3 1 0 0 0 0 1 0 1 0 = E I = 1 0 0 0 1 0 0 0 1 7 R 1 7 0 0 0 1 0 0 0 1 = F I = 1 0 0 0 1 0 0 0 1 4 R 2 + R 3 1 0 0 0 1 0 0 4 1 = G Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 1.6 Elementary Matrices All 2 × 2 Elementary Matrices Elementary matrices can be any size, but most examples we shall use will be 2 × 2 or 3 × 3 . Here is a complete list of all possible 2 × 2 elementary matrices: 0 1 1 0 , a 0 0 1 , 1 0 0 a , 1 b 0 1 , 1 0 b 1 where a = 0 . Can you identify which elementary operation was used to produce each of these matrices? Is I itself an elementary matrix? Note that each of these matrices is invertible, and that its inverse is another elementary matrix of the same type: 0 1 1 0 , 1 / a 0 0 1 , 1 0 0 1 / a , 1 - b 0 1 , 1 0 - b 1 More than that, the inverses simply “undo” the original operation. Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla
Chapter 1 1.6 Elementary Matrices Significance of Elementary Matrices Let E be an elementary matrix obtained from I by applying a specific elementary operation to I . Then EA is the same as the

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section16 - Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1...

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