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Test+4+solution - MATH 17100 28446 Test 4 solutions 12/02...

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MATH 17100 28446 Test 4 solutions 12/02/ 2009 1. Consider the system of equations 1 2 3 4 1 2 3 4 2 4 4 1 2 4 2 x x x x x x x x , a) Rewriting the system in matrix form Ax b , defining the new quantities involved. 1 2 3 4 2 1 4 4 1 1 2 1 4 2 x x x x A , x , b b) Find the row-reduced echelon form of the augmented matrix for the system. 4 / 3 4 / 3 4 / 3 2 1 4 4 1 1 2 1 4 2 1 2 1 4 2 1 0 3 0 | 1 2 1 4 2 0 3 6 4 3 0 1 2 1 0 1 2 1 A b c) What is the rank of the coefficient matrix? What is the rank of the augmented matrix? Rank of A is 2 since the echelon form of the coefficient matrix has two non-zero rows Rank of | A b is also 2 since the echelon form of the augmented matrix also has two non- zero rows d) Identify the basic variables. Basic variables are identified by the pivot elements, they are 1 2 and x x . e) What is the general solution to the system? Free variables 3 4 and x x can take on arbitrary values: 3 4 and , , arbitrary x x   1 3 4 2 3 4 4 4 4 4 3 3 and 1 2 1 2 3 3 3 3 x x x x x x General solution is
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1 2 3 4 4 / 3 4 / 3 4 3 3 0 3 4 2 1 1 2 3 1 0 0 0 1 0 x x x x x , , arbitrary   (20 pts) 2. The augmented matrix for a system of equations in variables 1 2 3 4 5 6 7 , , , , , , x x x x x x x has the following row-reduced echelon form 0 1 0 0 3 0 7 2 0 0 0 1 5 0 6 8 0 0 0 0 0 1 4 9 Write out the general solution of the system
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