Midtermsol2002

# Midtermsol2002 - A Solution of Math 115 2002 Midterm Test...

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A Solution of Math 115 2002 Midterm Test Problem 1. (a) Determine the values of a and b for which the following system of linear equations has (i) exactly one solution, (ii) inFnitely many solution, and (iii) no solutions. x 1 + ax 3 =2 x 1 + x 2 +( a +1) x 3 =7 x 1 + x 2 +2 ax 3 = b . (b) ±or the values of a and b in part (ii) above, give the general solution to the system. Solution. (a) ( A ± ± b ) = 10 a 11 a +1 11 2 a ± ± ± ± ± ± 2 7 b R 2 R 2 R 1 R 3 R 3 R 1 / a 011 01 a ± ± ± ± ± ± 2 5 b R 3 R 3 R 2 / a 1 00 a 1 ± ± ± ± ± ± 2 5 b 5 Then (i) the system has exactly one solution when a 6 =1and b R . (ii) the system has inFnitely many solutions when a b =5 (iii) the system has no solution when a = 1 but b 6 =5. (b) When a =1, b =5,wehave 101 000 ± ± ± ± ± ± 2 5 0 Therefore x 1 + x 3 x 2 x 3 . Let x 3 = s .Th en x 1 s, x 2 s, x 3 = s and x 1 x 2 x 3 = 2 s 5 s s = s 1 1 1 + 2 5 0 . Problem 2. (a) A 3 × 3matr ix B has the following elementary row operations performed on it, in the order given: 1) R 1 R 1 + aR 2 (add a t imesrow2torow1) 2) R 3 ± R 2 ( interchange row 2 and row 3) 3) R 2 bR 2 (multiply row 2 by b ) where a and b are non-zero real numbers. ±ind the matrix A such that the matrix product AB gives the same result as preforming the above three elementary row operations on B .

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## This note was uploaded on 09/17/2011 for the course MATH 115 taught by Professor Dunbar during the Fall '07 term at Waterloo.

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Midtermsol2002 - A Solution of Math 115 2002 Midterm Test...

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