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Solution_1 - THE CHINESE UNIVERSITY OF HONG KONG Department...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications (Fall 2007) Homework 1 Solutions 1. (a) x +2 y +3 z = 8 x +3 y +3 z = 10 x +2 y +4 z = 9 The augmented matrix of this system is 1 2 3 8 1 3 3 10 1 2 4 9 (-1) times the first row was added to its second row, then (-1) times the first row was added to its third row. 1 2 3 8 0 1 0 2 0 0 1 1 Hence y = 2 z = 1 Substituting y = 2 , z = 1 into x + 2 y + 3 z = 8, we have x = 1. Thus, the solution is ( x, y, z ) = (1 , 2 , 1). (b) x + 2 y + 3 z = 0 4 x + 5 y + 6 z = 0 7 x + 8 y + 10 z = 0 The augmented matrix of this system is 1 2 3 0 4 5 6 0 7 8 10 0 (-4) times the first row was added to the second row. (-7) times the first row was added to the third row. 1 2 3 0 0 - 3 - 6 0 0 - 6 - 11 0 (-2) times the second row was added to the third row.
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2 1 2 3 0 0 - 3 - 6 0 0 0 1 0 z : z = 0 y : - 3 y - 6 z = 0 y = 0 x 1 : x + 2 y + 3 z = 0 x = 0 Thus, the solution is ( x, y, z ) = (0 , 0 , 0) 2. 2 A + B = 2 4 6 8 + 0 - 1 5 - 2 = 2 3 11 6 Similarly we obtain A - 3 B = 1 5 - 12 10 Hence (2 A + B )( A - 3 B ) = 2 3 11 6 ‚ • 1 5 - 12 10 = - 34 40 - 61 115 On the other hand, A 2 = 7 10 15 22 AB = 10 - 5 20 - 11 B 2 = - 15 2 - 10 - 1 Hence we obtain 2 A 2 - 5 AB - 3 B 2 = 14 20 30 44 - 50 - 25 100 - 55 - - 15 6 - 30 - 3 = - 21 39 - 40 102 The correct expansion of (2 A + B )( A - 3 B ) is (2 A + B )( A - 3 B ) = 2 A 2 - 6 AB + BA - 3 B 2 . 3. (-2) times the first row was added to the second row. (3) times the first row was added to the third row. 2 5 - 3 a 0 - 3 2 - 2 a + b 0 12 - 8 3 a + c (4) times the second row was added to the third row. 2 5 - 3 a 0 - 3 2 - 2 a + b 0 0 0 - 5 a + 4 b + c Thus, in order to make the augmented matrix correspond to a consistent system,we con- clude - 5 a + 4 b + c = 0.
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3 4. (a)The 4 × 8 matrix [ A | I 4 ] is 3 4 1 - 1 1 0 0 0 4 - 1 3 5 0 1 0 0 1 5 - 4 - 6 0 0 1 0 4 - 3 2 - 2 0 0 0 1 The first row was multiplied by (1/3),we obtain 1 4 / 3 1 / 3 - 1 / 3 1 / 3 0 0 0 4 - 1 3 5 0 1 0 0 1 5 - 4 - 6 0 0 1 0 4 - 3 2 - 2 0 0 0
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