Question

4. (a) (8 points) Find the general solution of x' = Ax, for A= 2. Write the solution in vector form. 1-1 -3 (b) (4 points) Using your vector solution, write a matrix solution X(t). (c) (4 points) Using the matrix solution from part (b), determine en

Answer 1

[-3 J 6 @ find the general solution x'= Axe, for A=1-1 7 . I write the solution in vector form 6 Using vector solution, write a matoia Solution 1 xht) O using the matoix Sozlution from part 6 , determine e Given A= - I 1-1-31 first find eigenvalues IA- 111=0 omine ett. co -3-1) -1-1 -1-0 1-ol -3 -11 / L-1 (x-2)(-3-*)-(-1)=0 3+2+3x+12+1=0 12+40+4 = 0 12+21+21+4 = 0 X(X+2) +2 (4+2) ao C+2) (x+2) = 0 x+2)2 = 0 =) da-2,-2. = -2 with multiplicity of 2 find eigen vectors (AI) If x= -2 61 _)-629669) -3)+(20()

ty To Brive (4) --) (3)=(3) reduce the matrix -6.) R38*R* loo The system associated with the eigen value 1=-2. (1123) (6) -66%) 6)-(3) This reduces to the equation x+y =0 =)2 = -4. puty-(7) - 15 18to, Let y=1 eigen v peet eigen vector (os) = ('). -t/all a t ate liv act) =ce . The soclution general