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Unformatted text preview: 1 1.1 SOLUTIONS Notes : The key exercises are 7 (or 11 or 12), 1922, and 25. For brevity, the symbols R1, R2,, stand for row 1 (or equation 1), row 2 (or equation 2), and so on. Additional notes are at the end of the section. 1 . 1 2 1 2 5 7 2 7 5 x x x x + = = 1 5 7 2 7 5 Replace R2 by R2 + (2)R1 and obtain: 1 2 2 5 7 3 9 x x x + = = 1 5 7 3 9 Scale R2 by 1/3: 1 2 2 5 7 3 x x x + = = 1 5 7 1 3 Replace R1 by R1 + (5)R2: 1 2 8 3 x x = = 1 8 1 3 The solution is ( x 1 , x 2 ) = (8, 3), or simply (8, 3). 2 . 1 2 1 2 2 4 4 5 7 1 1 x x x x + = + = 2 4 4 5 7 11 Scale R1 by 1/2 and obtain: 1 2 1 2 2 2 5 7 1 1 x x x x + = + = 1 2 2 5 7 11 Replace R2 by R2 + (5)R1: 1 2 2 2 2 3 2 1 x x x + = = 1 2 2 3 2 1 Scale R2 by 1/3: 1 2 2 2 2 7 x x x + = = 1 2 2 1 7 Replace R1 by R1 + (2)R2: 1 2 12 7 x x = = 1 12 1 7 The solution is ( x 1 , x 2 ) = (12, 7), or simply (12, 7). 2 CHAPTER 1 Linear Equations in Linear Algebra 3 . The point of intersection satisfies the system of two linear equations: 1 2 1 2 5 7 2 2 x x x x + = = 1 5 7 1 2 2 Replace R2 by R2 + (1)R1 and obtain: 1 2 2 5 7 7 9 x x x + = = 1 5 7 7 9 Scale R2 by 1/7: 1 2 2 5 7 9/7 x x x + = = 1 5 7 1 9/7 Replace R1 by R1 + (5)R2: 1 2 4/7 9/7 x x = = 1 4/7 1 9/7 The point of intersection is ( x 1 , x 2 ) = (4/7, 9/7). 4 . The point of intersection satisfies the system of two linear equations: 1 2 1 2 5 1 3 7 5 x x x x = = 1 5 1 3 7 5 Replace R2 by R2 + (3)R1 and obtain: 1 2 2 5 1 8 2 x x x = = 1 5 1 8 2 Scale R2 by 1/8: 1 2 2 5 1 1/4 x x x = = 1 5 1 1 1 / 4 Replace R1 by R1 + (5)R2: 1 2 9/4 1/4 x x = = 1 9/4 1 1/4 The point of intersection is ( x 1 , x 2 ) = (9/4, 1/4). 5 . The system is already in triangular form. The fourth equation is x 4 = 5, and the other equations do not contain the variable x 4 . The next two steps should be to use the variable x 3 in the third equation to eliminate that variable from the first two equations. In matrix notation, that means to replace R2 by its sum with 3 times R3, and then replace R1 by its sum with 5 times R3. 6 . One more step will put the system in triangular form. Replace R4 by its sum with 3 times R3, which produces 1 6 4 1 2 7 4 1 2 3 5 1 5 . After that, the next step is to scale the fourth row by 1/5....
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 Fall '08
 PAVLOVIC
 Linear Algebra, Algebra, Addition

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