Question

Question 5. (20 pts.) a)Find Eigenvalues and Eigenvectors of the following matrix 3 2 6 1 -2 3 L6 4 12 b) Find the Fourier series representation of the function with period 21 given by t2 0 <t<TE i < t < 270 f(t) = {.

Answer 1

& Given Matrix Az 3 li 6 3 2 4 12 The Here characteristic quation of 2-3? (to(A))+ ( Aut Aset A22) - det(A)=0 tulA): 3-2 +12-13 3 3 3 6 Ano:1 -8 ܝ ܟ A 33 li 6 2 All : 12 < 36 det (A): 3 6 -2 12 6 3 6 3 -2 6 12 R2 = 1/2 R3 o( since last and god fowe are identic an]

detal=0 The characteristic equation is 13-2274 (-36 +0-8)+0 : 0 3 - Qu) : 0 À (x²-3 00, 1-0, 1:131 Visty.un 13 1 18.57 (correct up to de cimal places] $15.7851 -9.785, O 15.785, 2.785 (correct up to 3 decimal places] Valves are Hence the eigen be the eigen Vector corresponding to Now let VIE eigen value - Then the x=0 (A-01)V,=0 -> Av, :0 3 96 Ti [ :0 1 ( 6 12 Then we have 3xq+212 +673 = 6 90,- 2X2 +37350 6xq futz + 12X3 = 0 sno - ->0

reduced an System ① can be (2) 3X1+ 2X2 + 6 x 2:0 OYL. +372:0 La Now 2 system Can reduron on Hi-N2 + 373-0 8x2 -3x3-0 a } Here 13 is free variable. Let, X=8 then from system 8x2-3x3-0 812 313 - : 3.8 X2-3 2 = 212-3x 2.3.-3.8 18 (x,-18

tence Vi= oow to . dequired is the ) Vector eigen . be the eigen corresponding to the het yi vector 3. 31 eigen value Valve Nqs 15.785 Then (A-15.785 I) V₂ = 0 12.785 y, 6 -17.785 3 u -3.785 11 42 G '93 Then we have 1- -12-785 y, + Qy2 + 693-0 y. 17.78592+ 39 3-0 system o can be reduced an yi-17-785 Ya +3y3 = 0 225.38132 + 14.35543 -0 }

from system cm @ we redared the system on -17.7859, 393 :D Y. 0.19793-0 Ja is free variant set 19-1 13:! 4 0.1979, Y,= 17.7859 - 393 =117.785):10.199)-3(1) = 0.504 ccorrect up to 3 decimal placa) Then 0.500 0.197 is the required eigen vector to the Now Let Uz u Iz be the eigen eigen vector corresponding eigen vare A = d.785 Then (A+2.785I) V3 = 0 5.785 2 6 11 ܕܠ :0 0.785 ما 13 6 u 14.785 Thon We have 5.7854, +972 +6 13:0 + 0.7453,4 345-0 . 30 ہ > system o reduced to cl to. 785c2 + 3 d3= 6. -2.5u1u, -11.35512=0 42 tu.46913=0 I is free variable letlug = 1 2.-4.469 dis'-0.785 12 -3.03 Here 0.508 0.508 4.469 Vza (. ] is othe grequired eigen vector corresponding to the en eigenvalue -2.785
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b Bilen-function f(t): +, Ost TT Then the fourier pieries representation of the TiON function.fi CUVIIN f(4) [ ( د ) 110 ( ! )) ודק. - Cather آلی (1) U 77 Y3 :1 TV 1x dx (6) n TI w/= |- |- an : T1 3 211 Ani + [ f(t) cosnt of 0 27 T : # $"t cosnt of ++ o. cosnt cH + [tt samt 1 - 1 At se poate or] 18 - $ cosnt at -[It count ☆ Isinn+18) + (0+ (1+ 1] count D 2 ²T 1. Corni 1 2 2 (-on . 9 AL (ov) ? bn I Set f(t) sin n7 dt 0 t so IT t2 sinnt at to . -t [1th rosnt 15 - 60" 2t count d+] a Camionnet-3 (it segment 15 - I spatot) ToI count + TC-09

E 네에 + [com ㅠn3 + le ㅠn3 bn = -1) (7) [t] Then the required becomes + 30 ( S (-1)^ (cset + (OSA+ + T C-10 Lision fourier series 푸에역 ㅠn3 하 Thank you .