Question

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Find the eigenvalues and corresponding eigenvectors for the matrix if the characteristic equation of the 4 -4 4 09-4 matrix is if the characteristic equation of the matrix is 23 – 1922 + 1102 – 200 = 0. 0-1 6

Answer 1

Sow ا Let A = [ ا ا - ا- 4 0 ی - | 0 To find the charchenstic eq Νσω Set (A-113 |=0 -ا ا =) ہوا۔ ها - (-4 O = 0 0 --| 6- 4-) -) = 0 اے | «ا) - 6 = ابا۔ (س) ( } (۸ه) * د/ رده) (68)

→ 4-1){54-91-68 +2243-0 74-1){50-198+1?} = 0 (4-1) { x²_15X +50% =0 714»){ (-5)(1–00)} =0 O' = (4-1) (X-5) (1-10)=0 => 1 x=4,5, 10 are thé eigenvalues so 1-4, 1-5, 1-10 Next to fend eigenvectore FOR X=4 Consider martruix [A-11] 4 1-1 -4 9-X 4 = 0 -4 4 5 -4 0 (.: x=4) 0 -12

o (Troue row echelon form) O Now set [A-AI ]x = 0 13] o | 0 0 0 0 ㅋ [ 4 212 23 x2 => 23 1- [8] O [X3=0 X2=0 is absent, so it is a free variable Since 4 so Let Tag = t Hence the eigen vector corresponding x=4 x= (7:)-(5)-(3) (:) Hence is the eigen vecto Corresponding = 4

Now for A=5 consider thé matrees [A-11] = [ 4-4 -4 9-1-4 0 6-7 -1 0 -4 u 4 -4 O 0 O 0 - ) frow echelon form r[ - то Now Bet a=0 [A-AI] 213 04 - 1 => X2-23 0 X2=X3 => 124=0], Let [x3=t] [22=t/ A=5 so 0 24 22 203 () + eigen veeter correspingo XG=( (8) 1=5 is eigen vector corresponding

Finally for d=10 Considen matres 9 24 0 [ A-L] - -- - 4 연 1 0 - 4 ५ (이 4 개 1 0 0뿔 CI [ ] row echelon form Now set 0 10 0 북 M 01. [8] ㅋ 니 23 이 0 0 O 개불적 0 지 2 th23 } 0 개 블정, 로 t2g 0 키 M 10 t 1122 = ut Let [x3=t Vectore corresponding So the eigen to eigenvale A=10 블 t 3 C X,o=( 불 -ut 개 2 니 t