Question

2. (-/20 Points] DETAILS POOLELINALG4 4.1.012. Show that a is an eigenvalue of A and find one eigenvector v corresponding to this eigenvalue. 61 - 1,2 = 5 A= 1 4 4 2 3 V =

Answer 1

00 SP Matrix na 1 2 3. and we have show that das is eigenvalue of find eigenvector а си matris A Con responding to ds. for eigenvalue, JA-AI | >> |0] J-45 1 2 4 (6-1) 1 (4-1) 1 2- 4 2 (3-1) 80 W +(-1) O 4 (3-1) Eveband alonj first (6-1) 147.000 200.00 (47 5) (6-4) (4-1) (3-2) – 2] -(3-1-43 -1 [ 2 - 4(4-1)] = 0 3) (6-4) (12-41-34+8-23-[-1-1] - (2-16+41] =0 (6-1) [P-4+13] +(1+1] - [14 – 193= 3) 612-421460-13 +7,12-101 +4+1-41414 =0

U - ß +13 ² – 551 +7520 13 -1373 +354-7570 which is characterstic equation put d=5 we have 33_13x (6) ² - 5585-7520 of matrix A. > 125 -325 +275 - 75 o to o 29 Here, dos Satisfy equation is effenvalue of matrix A. Then, das Also, we have to find eigen vector corresponding to das for eigenvector ( [A-117 V=0) ie. [A-11 zh az 24 -- JO sy 4 2 ) a, te - h se-t az 421 +222 - 2x 1-03 Compair both sides we have

D x, the- 89,- += $ 42, 72722N3 =0 By Crammers sule ince to find value of x, , 24 Ms. from Ⓡ.3 and take any or equation and sohre by Crammer's rule, we have x, +24 - az = 0 24 -3 +33 0 213 li 22 (1-1) (1+1) 33 H-1) Az O -2 Then, Corresponding to eigen value ei]См 1-5 vector is 24 de sz - [i] (after divieting by (-2).) which is eigen vector v Corresponding to eigen value das.

Q+601213(ziven, T! IR? – - IR is lineas transformation Such that T[?] - [***247 Here, we have to find [T]CCB where, BE =} (47(]} is basis of IR? 3153[!][:] And Now, 2x2 2. 94[?] = [8] - -{}}**{ } 8 311!] =( a + Bt8 Oox + 3tr 0.2 +0.ß +r 2 Compair both sides, we have X+ B4r=5 and Btr =-1 and 18=2 2 Bar = -1 put Then, B = -1-2 - B - Also, put r=2&B=-3 in x + 84 825 Then, X-3+2=5 =6

Then, T[!] = [ ] [14]}]+ 2[:] Also, 2 + 2XCD) OTO 3+ - (3) - "(!**[!]+ => 1 [3] =] +Br 0.2 + Btr 0.2 to.BY sides, we have Compair both & tß tro and ßt r = -2 and re- pret r=rl in B+ r = -2 Then, B = -2+1 --) - Also, in dt Btr-o put r=-18 ßat Then, Q+H+ = 0 > R=2 Then, T037 - [--[]+c»f}]++ [3] 6 2 Hence, Teca 2 -1

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