Exam3-test-solutions - MATH 415 / EXAM 3 test (1) Use...

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Unformatted text preview: MATH 415 / EXAM 3 test (1) Use Cramers Rule to determine x 1 . " 7 5 9 7 #" x 1 x 2 # = " 1- 1 # . x 1 = 1 5- 1 7 / 7 5 9 7 = 12 / 4 = 3 . The complete solution is x 1 = 3 ,x 2 =- 4 , which you can use to verify your answer, but the answer itself should use Cramers Rule. (2) Find the inverse matrix for A using the formula A- 1 = C T / det( A ). A = 1 1 0 2 1 1 4 2 1 C = - 1 2- 1 1 2 1- 1- 1 , C T = - 1- 1 1 2 1- 1 2- 1 . The product AC T = I . Thus det( A ) = 1 and A- 1 = C T . (3) Let A be a real symmetric matrix and let x and y be eigenvectors such that A x = x and A y = y for distinct eigenvalues and . Show that x and y are orthogonal. y T A x = y T x , x T A y = x T y . With A T = A the two left sides are equal. But then also the right sides are equal which is only posible for y T x = x T y = 0 ....
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This note was uploaded on 11/19/2010 for the course MATH 415 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Exam3-test-solutions - MATH 415 / EXAM 3 test (1) Use...

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