Solution_1 - THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications(Fall 2007 Homework 1 Solutions

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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.jeffchak-fuwong during the Spring '06 term at CUHK.

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Solution_1 - THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications(Fall 2007 Homework 1 Solutions

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