Math Biophysics, Fall 2011Homework 1(1) Read about finding eigenvalues using Maple. LetC=-4-33-332-3333-4366-65Use Maple to find a non-zero vectorXsuch thatCX= 2X.(2) Use Maple to find the inverses of the matricesM=123345356andN=22-i2 +i-2.(3) SupposeAis self adjoint and thatv1andv2are eigenvectors correspondingto distinct eigenvalues. Show thatv*1v2= 0. (First read the proof in theBrief Linear Algebra Review that eigenvalues of selfadjoint matrices arereal. Use a similar method of proof.)(4) IfAis a real 2×2 matrix such thatA0A=Iand detA= 1, show that forsomeθ,A=cosθ-sinθsinθcosθ.(5) LetRθbe the rotation matrixcosθ-sinθsinθcosθ.Show that(Rθ-I) (R-θ-I) = 2(1-cosθ)I.(6) Show that ifAis real and has real eigenvalueλ, then there is a real vector(a vector with real coordinates) which is an eigenvector corresponding to
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