University of Hong KongFall 2011Math 1813Mathematical Methods for Actuarial ScienceInstructor: Siye WuProblem Set 6(Optional1, due at 17:00, Friday, 2 December)I. LetA=3111113111113111113111113.(a) Show that 7 is an eigenvalue ofAwithout using the characteristic polynomial ofA.(b) Given thatαis the other eigenvalue ofAwith multiplicity 4, findαand its correspondingeigenspace.(c) WriteA= 2I5+uuT, whereu= (1,1,1,1,1)T. FindA−1by assuming that it is of the formβI5+γuuTfor someβ,γ∈R.II. Letf(x,y) =x4+y4+ 4kxyandg(x,y,z) =x+12y2+y+xz-12z2-32.(a) Find the value(s) ofkso that the surfacesz=f(x,y) andg(x,y,z) = 0 intersect orthogonallyat the point (1,0,1)T. [Note: Two surfaces intersect orthogonally at a point if their normal vectorsare perpendicular at that point.](b) In the above situation(s), find the equation of the plane through (1,0,1)Tthat is perpendicularto both surfaces.
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