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**Unformatted text preview: **MATH 20F WINTER 2007 MIDTERM EXAM I JANUARY 31 Problem 1: Consider the following system of linear equations x 1 + x 2 + 2 x 3 = 12 x 2 + x 3 = 5 3 x 1- 2 x 2 + x 3 = 11 . a). Write down the augmented matrix of the system, and use the row reduction algorithm to find a row echelon form of the matrix. Circle the pivot positions in the final matrix and list the pivot columns. Determine whether the system is consistent. If the system is consistent, find all solutions of the system. Write it in parametric vector form. b). Write the system in the form A x = b , with A being the coefficient matrix of the system. Determine if there exist c in R 3 such that the equation A x = c has no solution, and if there is such c , give an example. Describe the set of all c in R 3 for which A x = c does have a solution. Solution: 1a ). The augmented matrix is ¯ A = 1 1 2 12 1 1 5 3- 2 1 11 , and a row echelon form can be found by the following sequence of elementary row oper- ations: 1 1 2 12 1 1 5 3- 2 1 11 ∼ 1 1 2 12 1 1 5- 5- 5- 25 ∼ 1 1 2 12 1 1 5 0 0 0 . Here, the pivot positions are indicated by the boldface numbers. The pivot columns are the columns 1 and 2. (Note that since echelon form is not unique, the particular echelon form you obtained can be different than this, but the pivot positions should be the same.) Since the last column is not a pivot column, the system is consistent . We can per- form one more elementary row operation to obtain the reduced row echelon form of the augmented matrix: 1 1 2 12 1 1 5 0 0 0 ∼ 1 0 1 7 1 1 5 0 0 0 0 ....

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