Question

Let A be a 2x2 matrix with eigenvalues 4 and and corresponding eigenvectors V, = and v2 Let} be a solution of the difference equation X: 1 -AX. Xo' - a Computex, = Ax (Hint: You do not need to know itselt b. Find a formula for x, involving k and the eigenvectors V, and v2 a x Ax=(Type an integer or simplified fraction for each matrix element) b. xxv.v2 (Type expressions using k as the variable.)

Answer 1

Anslu Given thout Wer hat we a 2x2 maturin withe 'u and i eigenvalue "..=[4] and we 07 Vi uit 1943 . To slove teu san Xkti = An Xoa FRIT Ar uvi Aist a.-b at no = 8 () = aut by a-b-5 3-b=5 atb=1 -> a =3, -6=+2. b.--2 2a - 6. (5) 3(1) + (27) :-) 3.- 22 -lo (a) to compute ni= Ano. nezario = (34,-2 U₂) = 3 AUL-2 AY niz 3.40.- 2ų = 124 124 = 12[:]-(3) 12 atl => 25 12 --| 1 u 23 2

k and (b) To find the formula for mere involving tu eign vectors vectars ve and V: ber 21 3.4vi 21 tuule Ani = 3.4Au, -24 AU in = Ny 43 nu=Aug - = 3.4 U, -21 ur " 3.4 u,.- et ry nz. An=348, -21 3.4v, -21 - 2 que u continianing in this way me = 3.4", -24 Boyk) Yle u ak 21 2 V 뇨 uk