Question

If A= ( -1 2 -3. 4) then e^At

If A = C) 2th 2 4 then eat. = (3e (3et – 2e2t -2et + 2e2t) 3et – 3e2t -2et + 3e2t) de At What is · AeAt dt

Answer 1

(- 4 So detų = 3 2 - Riven H = Here siges Row Sum = 1 is an eigen value as A. We know that dum of eigen So x=1 value=Sum of diagonals two cice eigen values of A >> X2 -2 i.e x = 1 1 2 - 2 dince A has dist cigen value So A is diagonizable. So there is a non-singular matrein P and diagonal matton I s.d PAP-D or PDP?LA.

Al-Pelp eigen vector Come lo x,-1 i The clor of A(3) is an eigen Vector Coren to Fl. We know that its A is diagonizable Then exponential of Ais where = Pop- qind p Ay = x, y ; y= Y +242 3y, tuy -9, +2y = y => ay, -2y =0 ? Y = 42 and -34, +"ya => "Y, - 4 Y 2 = 0 Yoga had g=1 7 = 1 So y=(!) y -3 30) ( У,

La 2-0 2.0 Vector Corn to 2-2 an eigen So Az = lg 2 iz to G :):):C: 22 :) -2, +aza -32, tuta 22 -32, +42g = 242 - 3 2 -222 = 0 32=222 6 7, +233 +2%, > 34--23 = 0 and No ale have het 2 = 2 Zg = 3 Gji Verdon Conre ho 3:2 eigen Lel P- So 2= (3) Э so pt and =

3el zoet So al Potp- 3) ( :) :) 1 2 3 30t -zet ort 30t-reat 20t +2eat 4+382f -2 et +30

f So + Ž DR AB+ 31 We know that ex = 1 + 2 + 2 + 3 Al It Alt I (I + Ad + 2 Aga dt = A + d AR + = A (1 + 2A + 2203 ) doab + + So - Dost do at de AAL AAL AAL (ANS)