Question

Find all distinct eigenvalues of A. Then find the basic eigenvectors of A corresponding to each eigenvalue. For each eigenvalue, specify the number of basic eigenvectors corresponding to that eigenvalue, then enter the eigenvalue followed by the basic eigenvectors corresponding to that eigenvalue. [o -6 -61 A = 0 -7 -6 10 4 3 Number of distinct eigenvalues: 1 Number of Vectors: 1 C

Answer 1

A to -6 -67 " To -7 -6 Lo 4 3 To find characteristic equecetico, consider, det (A-2,5) = det / SA 6-679 10 -22 -6 50 Lo u 3-2) ie: 05--7-2)(3-2) +243 +660] - 660) orta (7+2) (3-2).. 247 O= 2(21-72+37-72) = -24 ²+212-247 -242 > 0 2 -2 - 4 -32 - 120 +42273 2201 is an characteristic equation = n(2²+40+3)=0 = a (2²+37+2+3)=0 » 26262+3)+(2+3)]=0 > a [c2+3)(2+1) 320 =) 10=o or 2203 or 2=-11 3 l 2. There are 3 distinct eigenvalues I say 2,20, 22=- 38 23=-1. To find eigen rectors: consider CA-1,5) v =0 i e. CA-OJOV=0 => ANZO je.ro -6 - lo 7 -6 / V2 = 10 - Lo u 3.Lv i ro i TV lo-7-6 1 v2 Lon 3 ] L vs ] 100

R27 Rztari rol irnir son R3 R3-4 RI . Lool LN3 - R3 - Rat R2 rollTvir 1:0 0 1 | 12 Looo] LV3] V is free variable of V2=0&. Vz+Voro. - Vg E-V25-030 '. Vanuit Ni 7 rin N2 Vi=1) 1. V=67-]] - 031) - For n=0, v= 1 (A+3I) VEO CO7 Nove For 22=-3 consider, CA - 728) Noie aj (3-6 -6TviT To-4 of 1 v2 Lo 4 6 ] Lug Rith Rif -2 -27. DE :] Minge )-[:] pot fotfel: 1101-13) Lon 6 L V3 R3 R3t R2 r1 -2 - 27 OO R2 A I Rzra2 - 27 win 10 1 312.11 Loo 0 JL - V3 is free variable of N2 + 2 V2 20 = V2 222 2 V) f Ni-2 V2-2V320 -) Vi2V2+2V3 = 262) 4372 Vz

:.---6---6-ad 11. for mans , v- -*® - Ni = -V3 ' Na VII (V3 7 Sali I v2 =1-31 1-312 . LV3 L N3 = VS . For 22=-3 nome, 28=-) consider, CA-231) Neo je CA +1.7) VEO - r -6 -67rvil ro Lo 4 4] LV R2 I R2 -6 Por otro -67 T-1:1 -on n L V3 R3 - R₂-4R2 re-6-6 TviT ro 0 1 | VR ELO Lo o o L v3 : Los 3Ng is free variable fi Vet V350 -->) Tec-V37 & V, -612-6V350 m Vi= 6 Vzt 6 V3 = 66-V33 +6 V3 20 v-ruit rot rol :.=)=43]-[i] = 6], 532) V3 0 1 :. For 23=-1, v= . Lid Ansuer: Number of distinct eigesvaines ; 3 Number of rectors: 3 0:40), - () -{1} -3