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e111k - e An undetermined homogeneous system must have at...

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MAS 3105 Jan 20, 2011 Quiz I and Key Prof. S. Hudson 1) Use Gaussian elimination to put the following system into reduced row echelon form. Use matrix notation. You don’t have to find the solution set. x 2 + x 3 =0 3 x 1 + 2 x 2 + x 3 =4 2) Label each system as underdetermined, overdetermined or square. Then describe how many solutions there are (maybe infinity!), and explain that briefly. A = 1 2 | 3 0 1 | 2 0 1 | 1 , B = 1 2 0 1 | 5 0 0 1 3 | 4 , C = 1 0 | 5 0 1 | 4 3) Answer each part with “True” or “False”. You don’t have to explain (but it doesn’t hurt, and might help if we decide later that a question was not totally clear). a) A square matrix in REF must have a lead one somewhere in the top row. b) A 3 by 4 matrix in RREF must have at least 9 zeroes. c) A 4 by 3 matrix in RREF must have at least 9 zeroes. d) Gaussian elimination can change an inconsistent system into a consistent one.
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Unformatted text preview: e) An undetermined homogeneous system must have at least two solutions. Remarks and Answers: This was not supposed to be very hard, and the average [based on the top 14 out of 18 grades] was high; 51/60, or 85%. The unofficial scale is A’s = 55-60 B’s = 49-54 C’s = 43-48 D’s = 37-42 F’s = 0-36 1) Start by swapping the rows (a Type I op): A = ± 1 0-1 / 3 | 4 / 3 0 1 1 | ² 2) A) Over; inconsistent ( x 2 = 1 = 2 is not possible); B) Under; infinitely-many solns (consistent with free variables); C) Square, a unique solution, (5,4). 3) FFTFT. I usually go over the TF in class after each quiz. You are welcome to ask more about these though. 1...
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