Question

(1 point) -1 -4 a. Given that V1 [ 2] and U2 --10 are eigenvectors of the matrix _2] determine the corresponding eigenvalues. 4 11 = 12 = = -4x b. Find the solution to the linear system of differential equations x' y' satisfying the initial conditions x(0) = -3 and y(0) = 4. 4x – 2y x(t) = y(t) =

Answer 1

* a. Given matrix be a -4 4 Starting from forming a new matrix by subtracting I from the diagonal entries of the given matrix -d-4 4 -|-2 0 Now We equation equal to o the determinant of the matrix -1-4 4. iela or org 82 + 6x + 8 4+2) +40 =0 13-2 or, 12 = -4 either when a-4 We have: [42]~[ ㄱ solve the matrix equation! [$][%]=[0] We take N2=2t then ==t so, ū t و مو [] = [3] [*] 1,3-4 the eisen vector eigen value corres Ponding to the When a = - -2 We have

[:]-[::] Now solve the matria equation: [28] [2]-[:] are take N2 at then vi Thus. I [&] = {i}t [i The eisen vector eisen corresponding to the value! d2 = 1-2 wi t2 = b. 2 Given system be. L x'=.-40 y's =42-24 a(o)=-3 y(o)=4 frow (C)(33) trow the eisen values of the matrin -4 0 4-2 be: d1=-4 and X2=-2 The The frow eisen vectors corresponding to the 2-4 b1.34-2) eisen vector's corresponding to A2=-2 be:2=( ) (3) .cehon to be the ) - ce-44 (-2) + (a) ₂ ov, / :) + De-zz +

so x wae 0 -ee-4t + De-27 x=-ce-ht or, and y = zce-4t toe-27 How Putting tao and x = -3 we have 3=-e.eo c. 3 and x = -3e-4€ Putting too and y = 4 we have: 4 = 2ctD or : 4 = 64 D >> D -- 2 soi y = r(t) = 6e-42 -26-22 -3e-47 6e-42 2e-2t y (t? -