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s1_f03_sol - Problem S1 Solutions 1.a x y 2 z = 1 x 4 y 2z...

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Problem S1 Solutions 1.a 2 1 4 2 5 0 2 1 4 2 5 4 2 5 2( ) 0 2 5 4 3 6 5 x y z x y z x y z x y z x y z x y z x y z x y x y + = − + + = + = + = − + + = + + = + = + = + = 2(3 6 5) 3(2 5 4) 3 2 2/3 3 6(2/3) 5 3 1 1/3 1/ 2 2/3 0 1 x y x y y y x x x z z + = + = = = + = = = + = = 1.b 1 1 2 1 1 4 2 5 . Subtract row 1 from row 2 1 1 1 0 1 1 2 1 0 3 4 6 . Subtract row 1 from row 3 1 1 1 0 1 1 2 1 0 3 4 6 0 0 1 1 1 2/3 1/3 z y x = = =
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1.c 1 5 2 5 4 2 0 1 1 1 1/3 1 1 2 3 1 4 2 1 1 1 1 1 2 1 5 2 1 0 1 2 2/3 1 1 2 3 1 4 2 1 1 1 1 1 1 1 4 5 1 1 0 3 1 1 1 2 3 1 4 2 1 1 1 det ( ) ( ) ( ) x x y y z z a b c d e f a ei fh b di fg c dh eg g h i = = = = = = = = = = +
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2.a 4 2 2 7 3 2 5 3 4 4 2 2 7 3 2 5 3 2 5 2( 3 4) 2 5 7 13 5( 2) 5 7 13 2 3 3/ 2 3/ 2 2 1/ 2 1/ 2 3(3/ 2) 4 1 x y z x y z x y z x y z x y z x y z x y z x y x y x y x y y y x x z z + + = + + = + = + + = + + = + + = + = + = + = + = + = = = + = = + = = 2.b 4 2 2 7 3 1 2 5 . Multiply row 1 by -3/4 and add to row 2 1 3 1 4 4 2 2 7 0 1/ 2 1/ 2 1/ 4 . Multiply row 1 by -1/4 and add to row 3 1 3 1 4 4 2 2 7 0 1/ 2 1/ 2 1/ 4 . Multiply row 2 by 5 and 0 5/ 2 3/ 2 9/ 4
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