11hw2 - Homework Assignment #2 8.35 Let the general 2 × 2...

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Unformatted text preview: Homework Assignment #2 8.35 Let the general 2 × 2 technology matrix be given by A = a b c d . Prove Theorem 8.13 directly for such a matrix using Theorem 8.8. Answer: Consider B = I- A . Then B = 1- a- b- c 1- d . By Theorem 8.8, this is invertible if and only if (1- a )(1- d )- bc 6 = 0. Since columns sum to something less than 1, 1- a > c and 1- d > b . Because b and c are positive, we multiply and conclude (1- a )(1- d ) > bc . Thus (1- a )(1- d )- bc > 0, so B is invertible. Also by Theorem 8.8, B- 1 = 1 (1- a )(1- d )- bc 1- d b c 1- a . The denominator is positive, and the entries are positive, so the matrix is positive. 9.12 Use Cramer’s rule to compute x 1 and x 2 in Example 9.4. Answer: The system in 9.4 is 1 1 1 12 2- 3 3 4 1 x 1 x 2 x 3 = 5- 4 . As in Example 9.4, the determinant of the coefficient matrix A is 35. We compute the determinants of B 1 = 0 1 1 5 2- 3 4 4 1 and B 2 = 1 1 12 5- 3 3 4 1...
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This note was uploaded on 01/01/2012 for the course ECO 7405 taught by Professor Staff during the Fall '08 term at FIU.

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11hw2 - Homework Assignment #2 8.35 Let the general 2 × 2...

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