Question

ts) Find three linearly independent characteristic vectors and all characteristic roots of the matrix 3 2 -3 -3 -4 9 1-1 - 5

Answer 1

Ket, A-3 2-37 -3 -4 9 | -1 - 25 The charge herinitio equabion in det (xIg-A) = 0 det (A-253) AG 13.d 2-3 | -3 -2-4 9 l = 0 - - 25-02 + -21+(.4)(5-1) (--4) +60 =0 * <200+ $45-52 -32 +82) (--4)346030 -202 + $ (15-84+22)61-4) } +60 =0 - 212+ - 152489²_23_60+322 –423 } +60 =0 + -212+142-19 +422-60+6020 + -13+422-42-0 + 7 (92-44+4) 20 + (7-2)2 20 azo, 2, 2. Ther, the characterintie roots are, 0, 2,2

ow For 2.0 A-RI3 = A-o. Iz A = 3 2 - 37 - 2 Alow, T3 2 -31, |-3-4 91 0-2 6 L-1 -2 5 R Rg' = Rg - 8 R2 Thein, oo o — Teod] R. 3031 Die I R = R, IR 3 2-37 10+1-3 o -26 0 -2 6 looo 125 Loool Lo e o T32- 37 17 01-3 [1-25 Moo, solving the matrix equation, FloTru, 1.3 u2 Looo Luz Us +63 20, 82-30 z zo Zef, og at (ay rcalert) Toy Vzot, 42 = 37 Thus, o=13 It DHE E

loce, for a=2 . And I3 = A-2I3 Tai 2 -3 ] -3 -6 9 L -1 -2 3 I. Ro=R2+3 R Mooi 1-3 L-1 2-3 -6 9 -2 3 3R, Tie2-3) Rzt R. در 1 1 2 -3 1 1 2 -37 2 L-1 -2 3 Looo Solving the matrice equation 2 -3 L:: E d : m to Utog -303 0. set, weat and Uzas Then, & = t3 saat 62at, ozis 135. 24T T- 21 J Therefore, | 0

All characteristic roots are 0, 2,2 characturintie and three veetorno independent ane dro to w ) L ا