Question

® Define TELive) T(X,Y,2)=(2x, 48452, 4y+32) by vectors a) Find its characteristic Polynomial b) use a) to find eigenvalues and eigen c) Diagonalize MCT) d) Find adj (MCT) e) Find inverse of MCT)

Answer 1

2 TEL ( ) T(4,2)= (2x, 4y +52, 4y +32) let VERS - The general basis & fon R3 is given by ={(1,09,10,60 , 0,671 TL 19,0) = (2,0,0) 2. © +0.0 + 0.03 T 01:0) (6, 4,4) = 0 € + 9 @z — 4.03 T(0,0,1)= T(0,5,3) = 0.8 + 5.82 $ 3.99 2 [Ta = A (sexy) 5 3 ... Its characteristic polynomial is given by JA-211=0 0 - (2-1){(4-1)(3 – ) -20)0 4-5 o 3_^

>> (2-1)((A-1)(3-4) - 20) (2-1)(12 12 - 48-31+ A?-20). ** -> -) (12.78-8) 222141-16 13 71? le capired choseosclericalic དུ ( b -23 +942. CX-16 50 x² qx²+ 6 x 2160) is the is the required) polynomial Row. the eigenvalues given by (22_912+6x + c) -> (2-1)(72-72-8)=0 =) AZ, 2-x)(A²-SA+A-S).to (2-1)( ^(4-5)+1(1-3). -> (2-x) ( @+i (1-8))=0 (2-0) (9 +1)(1-8)=0 2= 2, 8,-1 The are 17 =-,?,8 . The eigenvector xllcorresponding to neol is given by (A+1) X=0 o 4+15 eigenvalues 2+1 1 (3) Como 113 0 4 311 >) 3x, = 0 572+5713=0 40 + 4913=0 -) / X2 = ->3 xo 게 763 -13 Az A - Az is free 1 a at Hy = 71 ( 1 ) to low the eigenvector corresponding eigenvalue =2 is given by XR) 22 "" 0 0 025 041 (A - 23)X =0 =) 2x2 +583=0 922 + x3 =0 712 143 4-25 A 3-2 and ai is free 4802 + 10x3 = 0 Ax2 + x3 = lef 41=1 qaga Pl=0 Xooo

lsame, X (2) mei 12 2=8 is given by 8/3) How the 2-8 (A - 81) XRO eigenvector fest =) NI 4-8 555 4 3-8 713 600 - 6X, EO Do 0-4 5 0 4-5 22 73 and - 474+ 573 -0 4x2 - 5X3 =0 Both arp => x=0, Al2 = 583 R, 3-0 H2 = 5/X3 21 ,' x 13 (184) (*- xs is free) 22 4 5/423) = x3 X3 (3) x= 5/4 The eigenvectors are xl, x®) >*(3) be {0.(9,6} с alowa 808 Klisuse the makofeigenvalue is And Cleavily all the the eigenvectores que linearly Independente matrix of eigenvector 0 P = 514 Б alow The PH 4P 6 How it is given by det (P) = 0 -16-1-514) and adj' (P) is given by prl= 1 adj (P) detle! ala CA Fultz (-1-514) co C12 = 6) 43 = (ight3 1

C21 (1) 2+1 (1) (134 914 0 -1 E 3 SIA 5/A 1 o ya 419 D= 67 012- CD (0), 913 = 60'+3 C22 (+)2+2 (o), C23 = Ej2+3 (6) C31 (5/8) C32 = 60342 (0) C33=(1) Matrix of Cofactor is adje ol sp- % la 51a plAP = Ala sia 2 o -4+5 o 5+5 era 0419-9+3 0 5+3 5+3 o la sig 3279 Hence D = PHAP @ alow the cofactors of MLT) = A is given by GAY 2 o 419 sig) - SIA =( 10 45 4 3 419 Bod 1.419 sla 1 -40%+40 10 8 9/ania 2 va Ala 11 qa 2 = D 7219 ore A= PDP-1 142 (0) all (o't ($12-20) (-1)2+1 (0) a 21 922 = 923 = (-1) 2+2 (6) (-1037210) (-11213 (5) ,932 = (1, 3+3 (6) a 31 (-1)3+ (O) 932= (-1)3+21 8 ... Mamex of cofactor is 6 o -10 8

8 Ady (MIT)) 6 6 8