assign3_soln

assign3_soln - Math 136 Assignment 3 Solutions 1. For each...

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Math 136 Assignment 3 Solutions 1. For each of the following systems of linear equations: i) Write the augmented matrix. ii) Row-reduce the augmented matrix into row echelon form. iii) Find the general solution of the system or explain why the system is inconsistent. a) x 1 + x 2 = - 7 2 x 1 +4 x 2 + x 3 = - 16 x 1 +2 x 2 + x 3 =9 Solution: i) 110 - 7 241 - 16 121 9 . ii) - 7 - 16 9 r 2 - 2 r 1 r 3 - r 1 - 7 021 - 2 011 16 r 2 r 3 - 7 16 - 2 r 3 - 2 r 2 1 1 0 - 7 0 1 1 16 00 - 1 - 34 ( - 1) r 3 - 7 16 001 34 iii) By back-substitution we get x 3 = 34, x 2 = 16 - 34 = - 18 and x 1 = - 7 - ( - 18) = 11. Hence the general solution is ( x 1 ,x 2 3 ) = (11 , - 18 , 34). 1
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2 b) x 1 + x 2 +2 x 3 + x 4 =3 x 1 x 2 +4 x 3 + x 4 =7 x 1 + x 4 = - 21 Solution: i) 1121 3 1241 7 1001 - 21 . ii) 3 7 - 21 r 2 - r 1 r 3 - r 1 1 1 2 1 3 0 1 2 0 4 0 - 1 - 20 - 24 r 3 + r 2 3 0120 4 0000 - 20 ( - 1 20 ) r 3 3 4 1 iii) Since the last row in REF has the form [0 · · · 0 | 1] the system is inconsistent. c) (1 + i ) z 1 iz 2 =1 (1 + i ) z 2 + z 3 = 1 2 - 1 2 i z 1 - z 3 =0 Solution: i) 1+ i 2 i 0 1 0 1 + i 1 1 2 - 1 2 i 10 - 1 a .
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This note was uploaded on 04/30/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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assign3_soln - Math 136 Assignment 3 Solutions 1. For each...

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