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Exam3-test-solutions

# Exam3-test-solutions - MATH 415 EXAM 3 test(1 Use Cramers...

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MATH 415 / EXAM 3 test (1) Use Cramer’s 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 Cramer’s 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 0 - 1 1 2 1 - 1 - 1 , C T = - 1 - 1 1 2 1 - 1 0 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 = λ x and A y = μ y for distinct eigenvalues λ and μ . Show that x and y are orthogonal. y T A x = y T λ x , x T A y = x T μ 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 . (4) Show that the matrix A is positive definite and determine its Cholesky de- composition. A = 1 2 1 2 8 2 1 2 5 A = LDL T = 1 0 0 2 1 0 1 0 1 1 0 0 0 4 0 0 0 4

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

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