e103k

# e103k - a A 3x4 augmented matrix in RREF must have at least...

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MAS 3105 Jan 21, 2003 Quiz I 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 =1 2 x 1 + 2 x 2 + x 3 =4 Answer: Switch the 2 rows, divide the 1st by 2, and subtract the 2nd from the 1st: ± 1 0 - 1 / 2 | 1 0 1 1 | 1 ² 2) This augmented matrix is in RREF. Find the solution set, using α notation (if necessary) in your answer. ± 1 2 0 1 | 5 0 0 1 3 | 4 ² Answer: x 4 = α , x 3 = 4 - 3 α , x 2 = β , x 1 = 5 - α - 2 β . So, S = { (5 - α - 2 β,β, 4 - 3 α,α ) } . [It is a good idea to check your answer. For example, set α = 1 and β = 2 and check that (0,2,1,1) is a solution. This is more important in harder problems]. 3) Answer each part with “True” or “False”.
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Unformatted text preview: a) A 3x4 augmented matrix in RREF must have at least two leading 1’s . b) Gaussian elimination can change an inconsistent system into a consistent one. c) Any two inconsistent 2x5 systems are equivalent. d) An underdetermined system can be inconsistent. e) If AB = AC and A 6 = O (the zero matrix), then B = C . Answer: FFTTF SMALL BONUS: Justify your answer to the last True-False question. Answer: Give a speciﬁc counterexample using 2x2 matrices. There are many possibilities, but A has to be singular. For example, set A = ± 1 ² B = ± 1 ² C = ± 2 ² 1...
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