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Assignment Solutions_a6s - MATH 2050 Assignment 6 Fall 2013...

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MATH 2050 Assignment 6 Fall 2013 Solutions 1. a) 1 - 1 2 0 - 1 3 0 4 0 1 2 4 2 - 2 4 0 r 2 + r 1 -→ r 4 - 2 r 1 1 - 1 2 0 0 2 2 4 0 1 2 4 0 0 0 0 r 2 r 3 -→ 1 - 1 2 0 0 1 2 4 0 2 2 4 0 0 0 0 r 3 - 2 r 2 -→ 1 - 1 2 0 0 1 2 4 0 0 - 2 - 4 0 0 0 0 (The form is NOT unique!) b) So the pivots are 1,1,and -2 (Not unique). c) The pivots columns are the first three columns. d) The rank of A is 3. e) Based on the result in a), 1 - 1 2 0 - 1 3 0 4 0 1 2 4 2 - 2 4 0 -→ 1 - 1 2 0 0 1 2 4 0 0 - 2 - 4 0 0 0 0 - 1 2 r 3 -→ 1 - 1 2 0 0 1 2 4 0 0 1 2 0 0 0 0 r 1 + r 2 -→ 1 0 4 4 0 1 2 4 0 0 1 2 0 0 0 0 r 1 - 4 r 3 -→ r 2 - 2 r 3 1 0 0 - 4 0 1 0 0 0 0 1 2 0 0 0 0 . 2. a) 0 3 2 9 1 1 - 1 0 2 5 0 - 7 r 1 r 2 -→ 1 1 - 1 0 0 3 2 9 2 5 0 - 7 r 3 - 2 r 1 -→ 1 1 - 1 0 0 3 2 9 0 3 2 - 7 r 3 - r 2 -→ 1 1 - 1 0 0 3 2 9 0 0 0 - 16 , so the rank of A is R ( A ) = 3. b) A = - 1 1 2 1 3 1 - 1 1 3 - 4 - 8 8 7 - 4 36 r 2 + r 1 -→ r 3 - 8 r 1 - 1 1 2 1 3 0 0 3 4 - 1 0 0 - 9 - 12 12 -→ r 3 + 3 r 2 - 1 1 2 1 3 0 0 3 4 - 1 0 0 0 0 9 ,
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so the rank of A is R ( A ) = 3. (The pivot columns are columns 1, 3 and 5.) c) A = 1 3 - 4 - 1 - 1 1 2 0 1 - 2 4 a
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