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Unformatted text preview: Math 215 Spring 2003â€”HW #1 Â§ 1.1, #13: Solve the system x 1 3 x 3 = 8 2 x 1 + 2 x 2 + 9 x 3 = 7 x 2 + 5 x 3 = 2 We write the augmented matrix: 1 0 3 8 2 2 9 7 0 1 5 2 Now replace Row 2 by Row 2 plus 2 times Row 1: 1 0 3 8 0 2 15 9 0 1 5 2 Now replace Row 3 by Row 3 plus 1 / 2 times Row 2: 1 0 3 8 0 2 15 9 0 0 5 2 5 2 The last row translates into the equation 5 2 x 3 = 5 2 , so that x 3 = 1. The second row translates into the equation 2 x 2 + 15 x 3 = 9, or 2 x 2 15 = 9, or 2 x 2 = 6, or x 2 = 3. The first row translates into the equation x 1 3 x 3 = 8, or x 1 + 3 = 8, or x 1 = 5. So, ( x 1 , x 2 , x 3 ) = (5 , 3 , 1). Â§ 1.1, #16: Determine if the following system is consistent x 1 2 x 4 = 3 2 x 2 + 2 x 3 = 0 x 3 + 3 x 4 = 1 2 x 1 + 3 x 2 + 2 x 3 + x 4 = 5 We write the augmented matrix: 1 0 0 2 3 2 2 0 1 3 1 2 3 2 1 5 Now replace Row 4 by Row 4 plus 2 times Row 1: 1 0 0 2 3 0 2 2 0 0 1 3 1 0 3 2 3 1 Now replace Row 4 by Row 4 plus 3 / 2 times Row 2: 1 0 2 3 0 2 2 0 0 1 3 1 0 0 1 3 1 Now replace Row 4 by Row 4 plus Row 3: 1 0 0 2 3 0 2 2 0 0 1 3 1 0 0 0 We have reached row echelon form, and there are no pivots in the last column, so, yes, the system is consistent (True, this is a Â§ 1.2 method, but give me a break). We could let x 4 be anything we want, then use the third row to find x 3 , then the second row to find x 2 , and then the first row to find x 1 . Â§ 1.1, # 19: Determine the value(s) of h such that the following matrix is the augmented matrix of a consistent linear system: 1 h 4 3 6 8 We replace Row 2 by Row 2 + 3 times Row 1: 1 h 4 0 6 3 h 4 Now, if 6 3 h 6 = 0, then the last line says that (6 3 h ) x 2 = 4, so that x 2 = 4 6 3 h , and then the first line will allow us to solve for x 1 , so the system is consistent. On the other hand, if 6 3 h = 0, then the second line says 0 = 4, which is not true, so the system is inconsistent. Thus, the system is consistent if h 6 = 2. Â§ 1.1, # 28: Suppose a 6 = 0 and the system below is consistent for all possible values of f and g . What can you say about the coefficients a, b, c, d ? ax 1 + bx 2 = f cx 1 + dx 2 = g We write the augmented matrix: a b f c d g We replace Row 2 by Row 2 plus c/a times Row 1: a b f d cb/a g fc/a The only way for the system to be inconsistent is if the last row is [00nonzero]. Thus, as long as d cb/a 6 = 0, we know that the system is consistent. If d cb/a = 0, then there any many values of ( f, g ) for which the system is inconsistent, namely any f, g with g c/a 6 = 0. So, given what weâ€™re told, we can say that d cb/a 6 = 0, or, ad bc 6 = 0....
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This note was uploaded on 09/19/2009 for the course MATH 330 taught by Professor Johnson,j during the Spring '08 term at Nevada.
 Spring '08
 Johnson,J
 Math, Linear Algebra, Algebra

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