Question

Q3. Given the following matrices, A=[ 3), B =[10], C= [31 a. Find the characteristic polynomial of A, B, C respectively. b. Is A diagonalizable? Is B diagonalizable? Is C diagonalizable? If no, please state your reason. If yes, please find the matrix P and D such that p-1MP = D c. Is the matrix A similar to the matrix C? Please explain your answer briefly.

Answer 1

Sola 3) a) A: 1 3 - The characteristic polynomial of JA u given by blage det LA-AL) 11- 3L - 3 9-11 - =(1-2)(9-4) - 9 - 9-1-94 +1-8 | P(A) = ² - lod = (1-10) Now B- 10 0 i prise det (B-II) 110 - 0 40- = -x(10-2) - = - (10- Ip(d) = 2 -lox = x(x-10) , 27 vent, c= 2 4 The charactuistic polynomial of a ů geven ky bCA) = det (€-11) ci-d 2 2 4- d am (1-2) (4-0) - 4

pcd) = 4-d=4&t ² - 4 Ipa) = ² - 5d = (1-5) $|| Now, مع معلمه scopi find the eigenvalue Put pla)=0 =0, d = 10. 1 leigenvalley) find the eigenveciou. id=0 (A - OI) v=0 Ti-o 3 V]-[º! a +3 = 0 x 2 = -2 v = v. 1-302 -302 - 1-3 | L IVE VIO (met 4 =1) الملتيه (ند (A-10 IVO A11-10 3 Tur 5. 3 9-10 1-9 3 121. IV. .To -1 30 - 4 =0 / VE 11%

Here whe number of eigenecious مع للطللصفطلعليلمحول( A. of columns in So, mabix A diagonalizables e-[-3/NTO Vio No 1 o 1oo 3 / Vio o 10 such that PART=0

foi matix B sapil Ligenvalleri Put plazo - otro, de = lo stepu Ligenve clou. 11=0 (B- OI) v=0 Nysa [V 107-107 107 truth NOV V2 ü) 12 = 10 TB-ToIJV=0 T10-10 o TwoT 0-10 40 -1002=0 av = ₂4 vi - [5/7 = [5/ 7 (het i = 1) Number of eigenrectou say, 2 i mol to number of columns of b. so B in diagonali kakle

15 2155 SL.05 Sush that LPBP-I – DIE OP

- seep for matrix a figenatud p (d)=0 A(1-5)=0 Slepo Eigen vectors i) 1,= 0 CC-OI) V-0 +218 = 0 a v = -2 V V5 i ) = 5 11-5 21 1 1 2 4-5 || V O 15-4,2 Tvaror 19 = 20,- = 2 = 0 a = 72 2 het u=1) thue, matux Idiagonalizable c . ů also ?

P- 05/1290-100 27129 0 10 such that. Pept =D