Question

AX, Consider a certain system of lineer equations = where Ais a 2x2 matrix of real numbers. Suppose one of the agen uclues of the coefficient matrix A is r=2+1, which has a corresponding eigen vector (2) What is the systen's real-valued general solution?

Answer 1

Answer X'-AX Lef x=1* cond A = a b d at daz dt and 1=2ti Xi 1 The given ugenuduse and the ean expenfunction Saterfing Ax = x AX, = AXI (2+0) cd 29 + bi 4 + 20 2C+ de +2 2a = 4 = ا ) 120= -1 a=2 b=2 ) c = 1/2 d=2 A= -122

the च system X =AX gelven by ari at 2 11 doly at -492 n2 = 20 +202 at don at ni +212 Let ni = A emt be the soln a = Bemt Tegn (*) Substituting in (A) we get maent = 2Aemt t2Bemt mbent Laemt 2 Be mt MA - = 2A + DB MB = -A +2B 뿔 (M-2) A-2B (DA + (m-1B - o D

For m = 2ti egn has Non truinal soin y m-2 - 2 m-2 m-2)2 +1=0 m2_um +4 + = 0 m2 -4m +5=0 m = 4 z 5 16 = 42 4 - 4 tre - 220 Let The soin be 2 = Azsin e 2+ZA, lost ezt? B, cast Sint} Les 4 - B2 Sint and S ni = ezt ? A coast + Ai sent? areer PB2 Lost to, sont

Substlecting 6 in (re) will get e 2014 -Alsent-A2 Cast) et 2e4 (Al cost-Assent) (Alcast-A. Assunt) Bi cast -Besent) 2 ezt +2 zeth and est l-Bish-zct) treet (Blast - Besent) et (A lost -Az sent) (Bilast - Ba -Befort) + ze = (-A1-2A2) sint to A2 +241) Last (-24 - 2B2) sint + (x + 2B, ) Cart and (-B ) - 282) sinf + (-B2 + 2B, ) Cast ~2B2 sint + 2833) Sint + (-11 + 2B) cast +Az qualing both sides ull get -A1-2A2 --2A2-2B2 -A2 tot = ZAI+2B1 - B1-B2 - Az 2B2 2 -B₂+2B) AN +2B,

The General solution of eqn (a) green bey ssint) G ₂ au % = Geet (2 host + 2 sint) + Cheet (-2cast + ^2 = 4 est (west-sint) + Gent ( Coast tsint) where ubilarly constant ie 3(261-263)cast +(222 +262) Sint } ( 4 +62) cast + (-4 th) sint} G & ca au cubebung Constant ni - est whey