Question

Define TELive) T(X.7,2)=(2x, 44452, 4y +32) by c) Diagonalize MCT) d) Find adj (MCT)) e) Find inverse of MCT)

Answer 1

Ques:- define TELCU,U) by T(X, Y, Z) = (2x, 44+52 AY +32) (c) digonalize MCT) (d) find adj (MCTÍ (e) find inverse of MCT) TE LIV, U) that is T is a hinear operator from V-V de TV V Given T(X,Y,Z) = (2x, 4y +52, 4Y+32) written in matrix form T(X,Y,Z) = Ansi- 0 2 0 0 O 4 ( 117 - M.2 3 (a) diganalize MCT) M(T) = Now 2 0 O O 45 u 3 0 the di ago nol matrix of M = ptmp where p is the metrix formed by eigen vector of M. Mow we find eigenvertor of M Now characteristic Polynomial of M = 1M-771=0

2 2 ОО 0 us 0 43 0 0 0 0 1 0 2-1 0 0 u-l 5 4 37 (2-1) [(4-1) (3 +) -20] +0101+0[0] = = (2-1) [12-72498-20) = 0 ► 2-1) (98-7) -8 ) - 0 -> (2-1) (98-82+2-8) = 0 » (2-1) i A-8) (4+1) = 0 7=2, a Here Eigenvalue of Matrix Mi's P1=-1, P2 = 2 and Azad Now we find Eigenvector Correspon ding ito - (M-(-)I) U=0 (M +I) O 0 0 [ : ] 42 0 0 2 0 43 0 1 u 3 0 14 o 5 4 4 42 43 3 ly=0 52 + 54₂ 0 4ų + 43 20

solue (1) lii) & (i) weget Lyco Uy + 4z U = - 43 Moto Eigenvertor corresponding to Arzt is u = 42 Now we find Eigen veetor Correspon ding to A2 = 2 (M-2I) v 18703 18 O- 2 100 2 5 ou 0 0 o u 3 007 0 0 0 0 25 0 4 2 U₂ + 5 U₂ =0 4 0 ₂ + 3 =0 4 U2 +10 U₃ u U2+U₂ -0 gUz to Uz=0 -> T. [U2=0 and and [ U1 = €) (any arbitrary Constant / -U1=1 we choose r=1 Now Eigen vector comesponding to $2=2 U 3

Now eigen vector Corresponding to Pg=& (M-AI) W zo رقا 2 0 0 o us 0 43 olo 0 9 o-4 - 6W, = 0 (wis -4 Wy + 5 W₂=0 --(UV) 4 W₂ - 5W3=0 - (viii) solve (1) Nii), (ii) weget Lu w = 5W3 then Eigen vector Corresponding to Az=d is u then matrix p = ily or U2 wi Uz 02 U U3 W3 0 O 5. a 0 Now pl = let Adj p IP

1P1 - OL u matrix Transpose + [-5-4)= -9 Adj p = Co-factor matrx off we find Co-factor matrix for p a Now 1+1 = 0 11 = o u azz 093 az A23 5 1+2 -9 17 12 = (-1) u 0131 A 33 1+3 021 A22 A13=(-1) azz 4 = -4 1 0212 a 12 Azz 4 0 0 11 e 0.22 = 033 143 033 012 -1 (+) 2+2 031 041 0 2+3 (-0. A23 13 A31 932 -1 0 O 3+1 012 = 5 013 A 23 1 Azi (0) 1922 an 021 143 11 0 11 Azz (1) 342 -110 A23 313 011 012 -- 033 = (-) al 0 A22

Now Co-factor matrix of p Cuz aiz al A21 031 A23 A32 A33 0 0 -9 0 5 0 T 0 Adj p = 0 -4 O 0 0 کیا ؟ ou 5 0 Mow 0 -u pta 5 O 0 -9 0 then diagonal matirik D= pimp اند -5-9 2oo Tolo 0-4 5 los 9 0 9 -104 .04 3 10.2to. -4t50 0 -16+20 0-20+15 -10 toto 0 0 -4-4 0 10 05 -lo y 4 0 18 0 0 S 0 -8 -8 you '0+4+5 0 20-20 olatoto -18 toto 0.-1d to stou Loto-0 0 + 8.5 -0.4 0 -18 0 0 - 72 0 coll 9 O cel- 0

Now Diagonal matrix of M (T) = 919 01-9 % -9 01-9 -10 1-9 0-9 Lol-g of g -71-91 Diagonal matrix D of M(T) = O 0 0 2. 0 0 & Ques- (d) Find Adjoint MLT) Ans: - Adj M (T) = Transpose of Co-factor Matrix 'M(T). first we co-factor matrix c of MCT). of au a12 043 Now ME 2 0 A21 4 5 4 3 022 923 932 a 33 0 031 u A22 Azz 5 Mow G1= (-1))* = 12-20=-8 4 3 دی 1+2 0 G2 = (4) -1 S 3 0 azz A21 a31 Azi C131 A12 043 a 23 a33 A22 Azz 0 4 - 0 1+3 u G3 = (4) 0 1 -0. - 4 3 C4 = (-1) 2+1 a32 al 2 0 C22 = (-13242 033 043 A32 6 A31

C23 = (-1) 243 an al 2 2 0 5 A31 a 32 0 u 0 012 Qu3 A22 923 1 0 u 5 C39 = (-0341 (22 = (-1) 3+2 jan 013 2 O 1 -10 1921 A23 0 5 C33- 3+3 ay 012 a 22 1 8 J 0 1921 Now Co-factor matrix cofM= (GI G2 G3 C21 22 23 Lei C2 C33 0 0 -8 - 0 c = 0 6 0 o Mow Adj MCT) c T 0 - 0 0-100 0 6 -10 Adj M(T) 0 0 8 Ques El) inverse MCT) Anst- Inverse of MLT) ma Adj M IMI

Now JM1 = 2 0 0 0 4 5 4 3 IM 1 = -16 Invesse MLT) In Adjm= - 11 -doo -10 0 -8 8 01-16 0/16 0/16 01-16 61-16 -101-16 01-16 -81-16 01-16 2 0 -3/8 518 0 1/2 1/2 0 Adj A Al determinant of A Adja Trans fose of = 2[ 4x3 - 4x5] 2x-Q = -16 Inverse of MCT) Results - (A) diagonalize of MCT)(O)= (dd) Adjoint of MCT) (e) Inverse of MUIDE 318 579 1/2 - 1/2 Poiagonal matrix of A = P'AP where pis matrix formed by eigen vector of o o od = 6 -10 - 0 0 0 0 -8 '/ 0 0 O 0 TAI Thank you