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

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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 + uu T , where u = (1 , 1 , 1 , 1 , 1) T . Find A 1 by assuming that it is of the form βI 5 + γuu 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 , 0 , 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 , 0 , 1) T that is perpendicular to both surfaces.
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