Question

Problem 2. Find the eigenvalues Xi and the corresponding eigenvectors v; of the matrix -4 6 -12 A-3 -16, (3 3 8 and also find an invertible matrix P and a diagonal matrix D such that D=P-AP or A = PDP-

Answer 1

Answer: - * Given matrix 1-4 6 -127 13- 16k 13 -3 81 The characteristic equation of Å is Soon Tardil to where I = orol 1-4-0 6 -12 1 3 1-3 8= . = (-4-1) [61-4368-1) +18]-6[318-49-18] -12 1-9-3(-1-d3] =0 =(-4-1){-871-84+*+1876 [ 24-31-18 ] -12 [-9 +3+31320 - (-4-1) [ 12-74 +10] -6[~34+6] P12[31-6] = 0 (-uras e x² - 2x-5x+10] +18 (4-2) -36 (4-2)=0 = (-4-1) [ A(4-2)-5(1-2)] +(18-36)A-2)=0 (-u-x) cras) (x-2) * (-88) (x+2) = 0 (4-2) ( -utot20-12+54 -18] =0 (!-2) [ -x2 at t+2].20 = (1-2) (12-4-2) =0 * x- z zo cors X2_ -2 =0. ies to2 = 12 +1-24-2=0 :: Act1)^2 cdti)=0 i s (t+1) CX-2) o CamScanner lies do-1,2. lcs Scanned with

Therefore the eigen values are . To find the eigen vectors , we can write (A-AI) 4; = 0 where you say Casenis:- For a=2 - (A-2 I) y=0 5-4-2 6 -127 97 pol OOO lie [-6 6 -127 1 3 -3 6 l L 3 -3 6 [az get the system of equations -62, + 6% - 12*3=0 1. (ies x,-x2+2x3=0 to 3*, -3% + 62z=0 - X, - xy +2312 70 - solving O and we have X-42 +243 =0 cies one equation with three variastes. so Take nisk, and å ak2 khe de kilk, are arbitary constants. = sy = x + 2*3 - Vit 2012 mm = kit2 K2 kit2k2 Scanned with, CamScanner

vali and we = ( 4 ) are the eigen vectors Corresponding to eigen value *=2. (Ame2=G. Mis) caseit): For tan, then (A + I) Vę = 0 ſ ~4+1.6 3 -17I 3 -3 -12 6 8+ pa,7 || X (x ) 3 6 -127 , Then the L 3 -3 a long system of equations are . -34, + 6*2-1292=0 => X, -28+447 30 60 3*,+0.2176.43=0 1 xy + 2xz 30 3% 34, +994220 . : i-x2 + 3 x3 =0_ ģ ili solving O and @,0= -248+4*3=0 2x39 = 27- 2/, + 64no - + - -XI -2*2=0 - 31+2*=0 si 9,5-243 Here we have one equation with h two variables PS Scanledd wat haak al 213-2k konstant CamScanner

Ibers from © ) -2K -24_tuk 20' 22K L . - Ask -2k ? ..^= 1:]-()-3 - Vs=1+1 se lhe eigen vector corresponding to eigen value xan Now , From ", V and Vy we have a non singular matrix .p' such that prio-27 loi il The determinant.Ipl 1 0 -21. = 162-1-0-2 (1) = 1-2 = -1 #0 PI-| ? where adj(p) = [cofactors matric of p] To find the pla adi (p) برہم + - cofactors matrix = 1 ob PL I al + s 1 -3 to 14 -3 2 As Scanned with CamScanner

adjp = (cofactors matrix of PIT > ady P = n! -2 41 - 1 - 3 mi 2 I llow c U 1:1Pl-1) I al -3 2 ) w p 2 - 4 JM Mow, we have to show that D=PAP oksa! 2 -471-6 -127 1-1 1 -2 ] -3 $ ) - 1-2 4 -87 11 -1 2 Pr2 4 -81110-27 HOW, De PAP= 2 -2 6 2 52 0 0 7 n. De 2 0 0 loro is a diagonal matrix. e with Tuse
​​​​This problem is depending on the definitions of Eigen values,Eigen vectors and Diagonal matrix