Question

(1) Find the eigenvalues and corresponding eigenvectors of [o1 0 0 0 1 2 1 -2 HINT: Note that 13 + 2/2 - 1 - 2 can be regrouped as 1(12 - 1)+2(12-1). Then factor out the common (12 - 1). (2) Solve the equation Y" + 2y' - - 2y = 0) using the method of converting to a linear system of first-order ODE's. Show that the coefficient matrix is the 3 x 3 matrix from problem 1. Then find the system's solution using the eigenvectors and eigenvalues. At the very end, note that the vector solution has components for y, y',y". Thus the solution to the original ODE is just the first coordinate of your vector solution.

Slove 2nd problem plz

Answer 1

So Given 1 6 0 = A (let) -2 The Characteristics equation of A | A-211 = 0 + 0-2 I + 0 O -2 1 2 -2-7 --|-26-2-2)-1] - 1[0-2] + [o+22] = 0 X² (-2-2) + 2 + 2 = 0 - 22² - 23 +7 +220 - 3-222 +2+2=0 23 + 27²-2-2=0 (2-1) + 2(x²-1) = 0 (2+2) (2²-1) = 0 (2+2) (2-1) (2+1) =0 2=-2 t, -1 Eigen value for a=-2 2 0 0 2 1 X2 2 1 0 X3 R3 R3 - Ri

0 ㅗ 2 0 0 2x + x2 = 0 2x2 + x3 = 0 from 8 2 X2 = -2X1 Put x2 = -28, in equation 2 (-2x) + x3 =0 - 4 xi + x3 =0 4 x = - X2 X3 4 = k (let) 2 x =k X2 = -2K x3 = 4k - 2K Eigen vector 4K For a=1 - 1 1 - 1 1 JET 2 -3

R3 R3 +2 Ri 0 1 -3 R3 R3 + 3 R2 O 110 a X2 0 - x + x2zo X2 = x, - X2 + x3 zo => x3=X2 x = x2 = x3 =k Xt x2 =k X3 =k K R к → Eigen vectoy At 2:47 1 1 O O 1 1 10-13 11 2 1 R3 R3-2 R

[8 Х3 R3 R3 7 R2 EIN X3 X 2+x=0 x2 + x3 = 0 => x = -X2 = x2 = - X3 x = -X2 = = x3 = K (net) Xizk X2 =-K x3 = K K -k K