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Unformatted text preview: 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 4 5 6 7 8 10 • (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 3 6 6 11 • (2) times the second row was added to the third row. 2 1 2 3 3 6 1 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) 2. 2 A + B = • 2 4 6 8 ‚ + • 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 3 2 2 a + b 12 8 3 a + c • (4) times the second row was added to the third row. 2 5 3 a 3 2 2 a + b 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. 3 4. (a)The 4 × 8 matrix [ A  I 4 ] is 3 4 1 1 1 0 0 0...
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This note was uploaded on 05/18/2010 for the course MATHEMATIC MAT2310 taught by Professor Dr.jeffchakfuwong during the Spring '06 term at CUHK.
 Spring '06
 Dr.JeffChakFuWONG
 Linear Algebra, Algebra

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