Question

2 2 0 0 3" (12%) Solve the linear system x"(t) AX(t) with A 0 0 4 4 a) (406) Write down the characteristic equation of the coefficient matrix λ and solve its eigenvalues, . b) (8%) Find the four independent solutions of the system.

Answer 1

Given A= [1 0 0 0 2 2 0 0 0 3 3 0 0 0 4 4] given matrix is lower triangular matrix det (A-lambdaI) = 0 implies [1-lambda 0 0 0 2 2-lambda 0 0 0 3 3- lambda 0 0 0 4 4-lambda] = 0 (1- lambda)(2- lambda)(3- lambda)(4- lambda) notequalto 0 + 0 + 0 = 0 (2- lambda-2 lambda+ lambda^2)(3- lambda)(4- lambda) = 0 (2-3 lambda+ lambda^2)(3- lambda)(4- lambda) = 0 (6-2 lambda-9 lambda+ 3 lambda + 3 lambda^2- lambda^3) (4- lambda) = 0 (6-8 lambda+ 3 lambda^2- lambda^3) (4- lambda) = 0 24-6 lambda-8 lambda+ 8 lambda^2 + 12 lambda^2-3 lambda^3-4 lambda^3 + lambda^4 = 0 lambda^4-14 lambda+ 20 lambda^2-7 lambda^3 + 24 = 0 is characteristics polynomial and eigenvalue of A is (1, 2, 3, 4) (1 2 3 4) is independent solution of the given system.