ps6 - University of Hong Kong Fall 2011 Math 1813...

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Unformatted text preview: University of Hong Kong Fall 2011 Math 1813 Mathematical Methods for Actuarial Science Instructor: Siye Wu Problem Set 6 (Optional 1 , due at 17:00, Friday, 2 December) I. Let A =       3 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1 3       . (a) Show that 7 is an eigenvalue of A without using the characteristic polynomial of A . (b) Given that α is the other eigenvalue of A with multiplicity 4, find α and its corresponding eigenspace. (c) Write A = 2 I 5 + u u T , where u = (1 , 1 , 1 , 1 , 1) T . Find A − 1 by assuming that it is of the form βI 5 + γ u u T for some β, γ ∈ R . II. Let f ( x, y ) = x 4 + y 4 + 4 kxy and g ( x, y, z ) = x + 1 2 y 2 + y + xz- 1 2 z 2- 3 2 . (a) Find the value(s) of k so that the surfaces z = f ( x, y ) and g ( x, y, z ) = 0 intersect orthogonally at the point (1 , , 1) T . [Note: Two surfaces intersect orthogonally at a point if their normal vectors are perpendicular at that point.] (b) In the above situation(s), find the equation of the plane through (1...
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This note was uploaded on 12/20/2011 for the course MATH 1813 taught by Professor Wu during the Fall '11 term at HKU.

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