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Unformatted text preview: ASSIGNMENT 1 Solutions In questions 1 and 2, find all real values of a for which the system has: (i) exactly one solution; (ii) no solutions; and (iii) infinitely many solutions. 1. [2 marks] x 1 + 3 x 2 + 2 x 3 = 1 2 x 1 9 x 2 4 x 3 = 12 x 1 + 3 x 2 + ax 3 = a 2 23 The augmented matrix of the system,  1 3 2 1 2 9 4 12 1 3 a a 2 23 , reduces to the matrix  1 3 2 1 3 10 a + 2 ( a + 2) ( a 2) , whence the system has (i) exactly one solution if a 6 = 2 ; (ii) no solutions never; and (iii) infinitely many solutions if a = 2 . 2. [4 marks] ax 2 1 + x 2 2 = a 2 ( a 2 2 a 2 ) x 2 1 3 x 2 2 = a 4 2 a Let X 1 = x 2 1 and X 2 = x 2 2 , and consider the system X 2 + aX 1 = a 2 3 X 2 + ( a 2 2 a 2 ) X 1 = a 4 2 a. The augmented matrix reduces to the matrix M = " 1 a a 2 ( a + 2) ( a 1) a ( a + 2) ( a 1) 2 # . Suppose a = 0 . Then M reduces further to the matrix 1 1 hence x 2 1 = x 2 2 = 0 ; that is, x 1 = x 2 = 0 . Suppose a 6 = 0 . If a = 2 or a = 1 , M reduces to 1 2 4 or 1 1 1 , respectively. In the first case, we haverespectively....
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 Fall '08
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 Math

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