quiz1-sol

# quiz1-sol - | 0 0 1 3 | 1 0 0 0 |(Row echelon form R 2-2 R...

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Name: MATH227 Quiz 1 (Fall 2006) For full credit, please show all steps in details!! 1. For the following system of linear equations x 1 - 2 x 4 = - 3 2 x 2 + 2 x 3 = 0 x 3 + 3 x 4 = 1 - 2 x 1 + 3 x 2 + 2 x 3 + x 4 = 5 (a) Write down the coeﬃcient matrix and the augmented matrix. (2 points) (b) Reduce the augmented matrix to its row echelon form and reduced row ech- elon form . (5 points) (c) Find the general solutions of the system. (3 points) (a) Coeﬃcient matrix: 1 0 0 - 2 0 2 2 0 0 0 1 3 - 2 3 2 1 Augmented matrix: 1 0 0 - 2 | - 3 0 2 2 0 | 0 0 0 1 3 | 1 - 2 3 2 1 | 5 (b) 1 0 0 - 2 | - 3 0 2 2 0 | 0 0 0 1 3 | 1 - 2 3 2 1 | 5 R 4 +2 R 1 1 0 0 - 2 | - 3 0 2 2 0 | 0 0 0 1 3 | 1 0 3 2 - 3 | - 1 R 4 - 3 2 R 2 1 0 0 - 2 | - 3 0 2 2 0 | 0 0 0 1 3 | 1 0 0 - 1 - 3 | - 1 R 4 + R 3 1 0 0 - 2 | - 3 0 2 2
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Unformatted text preview: | 0 0 1 3 | 1 0 0 0 | (Row echelon form) R 2-2 R 3 ∼ 1 0 0-2 | -3 0 2 0-6 | -2 0 0 1 3 | 1 0 0 0 | 1 2 R 2 ∼ 1 0 0-2 | -3 0 1 0-3 | -1 0 0 1 3 | 1 0 0 0 | (Reduced row echelon form) (c) So the system of linear equations is equivalent to x 1-2 x 4 =-3 x 2-3 x 4 =-1 x 3 + 3 x 4 = 1 and the solution set is x 1 =-3 + 2 x 4 x 2 =-1 + 3 x 4 x 3 = 1-3 x 4 x 4 is free. 1...
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