Question

1. Find a general solution of X" = AX with -5 4 (1) A= (2) A= 4 -5 2 3 1 -2 4 (3) A= 2 2 3 -3 (4) A= -6 2 4

Answer 1

E A = -5 Now find eigenvalue and eigenvector of A - IA-211-0 1 1-5-d 4 16 = (-5-4) -1670 (4+5) = d+5 = 24 - 1 = -2, -1. - eigenvalues des-9, A2= -1 for, di --9 A-(9) = 11.11 be. :] Shocks. [...] to ] :26 +22=0 2 = ne 23:[*3]==217] eigenvector corresponding eigenualue diang in -=[:] for, d25-1 R2-4R, R2 lo:] [ A-(-VI : -[.] che, [4] 6:36.13 224 = 22 ny-n2-0 D] - [*] --2 [1]

: eigenvector corresponding eigenvalue dz=-1 is -L : The general solution. x = a é 1941;] et[] + с. е Ae [132] - - 2 Now find eigenvalue and eigenvedon of A SIA-20 2 12 1 -2- - 220 > (-3-1) (-2-1) - >$? +5d +6-2=0 ad² +50+4=0 > (8 + ) (a +1) =0 d=-9, -1 & eigenvalues are d=-4, d2 = -1 for 4 A-(-9) I -[: :7*** [...] (N-18 :24 +22₂ - 0 22 = -222 [*[] -. [13] eigenvector corresponding eigenvalue draaqis 21:3] 0,

4-6-1-(7 41 68. [ 17**** [ [1] [3]-[:] .24-22=0 2 = 2 [ma] [m] => [!] eigenvector corresponding eigenvalue diaries -- [1] .. The general solution. x=4&*[1] +5+*18 2 2 A- 3 Now find eigenvalue and eigenvector of A L-1A-Il=0 -2-2 2 3 (-2-2) (-3--) -6=0 otsd +6-620 >> 22 +5d=0 d=0,-5 3d (a +5)=0 & eigencealues are di = 0, de 3-5

diso -2 R₂-3 R₂ R2 276UR, A-(OI 325R [. :] 3 [B]=[-][- :) : mn2-0 22 =22 - [ ] [ ] = z2[:] eigenvector corresponding eigenvalue a, = 0 i [i] for d₂=-5 1 4-607-(3 :] bm.[: tip [ [ }] [21]+[:] n2 24+ 2222 = 0 are=-2/3 he (]. eigenvector corresponding eigenvalue gausis -13 .. The general solution ازه x=Ge "* [!] +62259[] *=*[9468*(13)

A= 2 Now find eigenvalue and eigenvector of A. ; IA-d11=0 1-6-2 4 2 -4-2 >(-6-4) (-4-22 -820 2 +10d +24-820 22 +10d + 16-0 -> (a +8) Ca +2) 20 »7--8,-2 are di=-8, d2=-2 & eigenealues dia-8 A-(-2) I - L: 1 [1.4-***.[:) [?] Q] =[:] 24 +222=0 24 = -222 1.1 [*] =2-13] eigenvector corresponding eigenvalue dia 8 is [3] for a2 = -2 R₂-2R1AR2 A-(-2) I - [2.2] Shri. (: ] [67] [3]:[:] 74- ho anna

N N (1) eigenvector corresponding eigenualue dz=r2 is 01 the general solution <=6€*[*3] +60=4[!]