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solutions10_2011

# solutions10_2011 - Math 152 Spring 2011 Solutions...

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Math 152, Spring 2011 Solutions Assignment #10 1. Consider the complex linear system: x 1 - 2 ix 2 = 3 i - x 1 + 3 x 2 = - 2 All complex numbers appearing in your final answers of the following questions must be in the form a + bi with a, b real numbers. (a) Write the system in the form A x = b where A is a matrix with complex entries and b is a vector with complex coordinates. (b) Compute det A . (c) Compute A - 1 . (d) Use the previous question to solve the original system. (e) What MATLAB commands would enter the matrix A and compute its inverse? Solution: (a) A = 1 - 2 i - 1 3 and b = 3 i - 2 . (b) det A = 1 · 3 - ( - 1) · ( - 2 i ) = 3 - 2 i . (c) Since det A = 3 - 2 i = 0, A is invertible and the formula for inverse of 2 × 2 matrices gives A - 1 = 1 3 - 2 i 3 2 i 1 1 = 3 + 2 i | 3 - 2 i | 2 3 2 i 1 1 = 3 + 2 i 13 3 2 i 1 1 and so A - 1 = 1 13 9 + 6 i - 4 + 6 i 3 + 2 i 3 + 2 i . 1

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(d) Since A x = b and A is invertible, we have a unique solution to the system: x = A - 1 b = 1 13 9 + 6 i - 4 + 6 i 3 + 2 i 3 + 2 i 3 i - 2 = 1 13 27 i - 18 + 8 - 12 i 9 i - 6 - 6 - 4 i = 1 13 - 10 + 15 i - 12 + 5 i , that is x 1 = - 10 13 + 15 13 i and x 2 = - 12 13 + 5 13 i . (e) A = [1 -2i; -1 3]; inv(A) or Aˆ(-1) 2. Let z 1 = 1 - i and z 2 = 1 + 3 i .
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solutions10_2011 - Math 152 Spring 2011 Solutions...

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